Correlated Gromov-Witten Invariants
We introduce a geometric refinement of Gromov-Witten invariants for ℙ¹-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a refinement of the degeneration formula, keeping track of the correlation....
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2025 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/213530 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Correlated Gromov-Witten Invariants. Thomas Blomme and Francesca Carocci. SIGMA 21 (2025), 046, 49 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860281479407337472 |
|---|---|
| author | Blomme, Thomas Carocci, Francesca |
| author_facet | Blomme, Thomas Carocci, Francesca |
| citation_txt | Correlated Gromov-Witten Invariants. Thomas Blomme and Francesca Carocci. SIGMA 21 (2025), 046, 49 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce a geometric refinement of Gromov-Witten invariants for ℙ¹-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a refinement of the degeneration formula, keeping track of the correlation. Finally, combining certain invariance properties of the correlated invariant, a local computation, and the refined degeneration formula, we follow floor diagram techniques to prove regularity results for the generating series of the invariants in the case of ℙ¹-bundles over elliptic curves. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces.
|
| first_indexed | 2026-03-17T20:39:16Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 046, 49 pages
Correlated Gromov–Witten Invariants
Thomas BLOMME a and Francesca CAROCCI b
a) Université de Neuchâtel, rue Émile Argan 11, Neuchâtel 2000, Switzerland
E-mail: thomas.blomme@unine.ch
b) Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma 00133, Italy
E-mail: carocci@mat.uniroma2.it
Received September 24, 2024, in final form June 09, 2025; Published online June 18, 2025
https://doi.org/10.3842/SIGMA.2025.046
Abstract. We introduce a geometric refinement of Gromov–Witten invariants for P1-
bundles relative to the natural fiberwise boundary structure. We call these refined invariant
correlated Gromov–Witten invariants. Furthermore, we prove a refinement of the degener-
ation formula keeping track of the correlation. Finally, combining certain invariance prop-
erties of the correlated invariant, a local computation and the refined degeneration formula
we follow floor diagram techniques to prove regularity results for the generating series of
the invariants in the case of P1-bundles over elliptic curves. Such invariants are expected to
play a role in the degeneration formula for reduced Gromov–Witten invariants for abelian
and K3 surfaces.
Key words: Gromov–Witten invariants; enumerative geometry; elliptic curves; decomposi-
tion formula
2020 Mathematics Subject Classification: 14N35; 14N10; 14J26
1 Introduction
1.1 Setting
In this paper, we introduce and study a refinement of the relative (or logarithmic) Gromov–
Witten invariants [1, 19, 24, 26] of the pair (Y = PX(O ⊕ L), D), where D = D+ + D− =
PX(L)⊕PX(O) and X is a smooth projective variety over C. We fix the following discrete data:
� g, n, m are positive integers,
� w = (w1, . . . , wn) is a n-tuple of non-zero integers with
∑
wi = 0; we moreover set
b(w) =
∑
wi>0wi = −
∑
wi<0wi;
� β ∈ H2(X,Z) is an effective homology class.
Let M := Mg,m
(
Y |D±, β,w
)
be the moduli space of logarithmic stable maps parametrizing
f : (C, p1, . . . , pm, q1, . . . , qn) → Y,
where C is a stable curve of genus g with image in class β + b(w)F ∈ H2(Y,Z), the marked
point qi is mapped to Dsgn(wi) with tangency order |wi|, the marked points pi are mapped
to the interior. The vector w is thus called tangency profile. The logarithmic (or equiva-
lently relative [2]) Gromov–Witten invariants are defined integrating constraints pulled-back
along evaluation maps ev : Mg,m
(
Y |D±, β,w
)
−→ Xn × Y m, against the virtual fundamental
class
[
M
]vir
[19].
We propose a refinement for these GW invariants by remembering a discrete quantity de-
pending on the position of the points mapped to the boundary divisor. This quantity is called
a correlator.
mailto:thomas.blomme@unine.ch
mailto:carocci@mat.uniroma2.it
https://doi.org/10.3842/SIGMA.2025.046
2 T. Blomme and F. Carocci
1.2 The Albanese evaluation
The idea of the refinement comes from reinterpreting the map to
(
Y,D±) = (PX(O ⊕ L), D±)
as the data of a map f : C → X together with the choice of an isomorphism f∗L ∼= OC(α)
where α ∈ H0
(
C,M
gp
C
)
is defined using the logarithmic structure on C. When C is smooth, this
condition takes the familiar form f∗L ≃ OC(
∑
wiqi).
1.2.1 Case of X × P1
To simplify, let us momentarily assume that L is the trivial bundle and that the source curve C
is smooth. Then, reinterpreting f ∈ M as explained above, part of the data of a log stable
map is the isomorphism OC
(∑
wiqi
) ∼= O ∈ Pic0(C). For a smooth curve, the Abel–Jacobi
theorem ensures that Pic0(C) ≃ Alb(C), where Alb(C) is the Albanese variety of C. We denote
by aX : X → Alb(X) the map from X to its Albanese variety, unique up to translation. By the
functoriality of the Albanese variety, f induces a morphism of Abelian varieties f∗ : Alb(C) →
Alb(X) compatible with the Albanese maps.
We denote by aw the morphism
aw : (x1, . . . , xn) ∈ Xn 7−→
n∑
i=1
wiaX(xi) ∈ Alb(X).
As
∑
wi = 0, the map aw does not change if we compose aX with a translation. From
OC(
∑
wiqi) ≃ O ∈ Pic0(C), we get that
n∑
i=1
wiaX(f(qi)) = 0 ∈ Alb(X). (1.1)
Equation (1.1) is a constraint on the relative position of the images f(qi). In other words, the Xn
part of the evaluation map is not surjective, and truly has values in the subset a−1
w (0).
1.2.2 Case of a non-trivial bundle
If L is not assumed to be trivial anymore, we prove in Lemma 3.2 that there is a natural map
φβ : Pic0(X) → Alb(X) defined through Hodge theory, which only depend on the curve class.
Then Proposition 3.4 ensures that equation (1.1) becomes the following:
n∑
i=1
wiaX(f(qi)) = φβ(L) ∈ Alb(X). (1.2)
In other words, the image depends only on the degree β and the line bundle L and not on f .
1.3 Correlated classes and correlated GW invariants
We now assume that the tangency orders wi have a non-trivial common divisor δ. We then
consider the map
aw/δ : (x1, . . . , xn) ∈ Xn 7−→
n∑
1
wi
δ
aX(xi) ∈ Alb(X).
The above morphism induces a morphism from the space of log stable maps M
κδ(f) =
∑ wi
δ
aX(f(qi)) ∈ Alb(X).
Correlated Gromov–Witten Invariants 3
It follows from equation (1.2) that δ · κδ(f) = φβ(L). Therefore, the map κδ takes values in the
set of δ-roots of φβ(L), denoted by T δ(L, β). The latter is a torsor under the group of δ-torsion
elements in Alb(X), denoted by Torδ(Alb(X)). The value of κδ is called a correlator and is
denoted by θ. We want to refine the Gromov–Witten of
(
Y,D±) keeping track of the value of
the correlator.
The discussion above assumes the source curve C to be smooth, but the morphism κδ is also
defined on the boundary of the moduli space Mg,m
(
Y |D±, β,w
)
, still with values in T δ(L, β)
(see Section 3.2.2). This shows that M splits into distinct (possibly still disconnected) compo-
nents Mθ indexed by the values of the correlator θ ∈ T δ(L, β); in particular the virtual class
splits as a sum of the so-called correlated classes
[
Mθ
]vir
. We define the full correlated class [[M]]δ
[[M]]δ =
∑
θ∈T δ(L,β)
[
Mθ
]vir · (θ) ∈ Q[Alb(X)]⊗H•(M,Q),
which is an element of the group algebra with cycle coefficients with support on T δ(L, β) ⊂
Alb(X).
Correlated GW invariants are obtained by capping these classes with some pullback by the
evaluation map: if γ ∈ H•(Xn × Y m,Q), we set
⟨⟨γ⟩⟩δg,β,w =
∫
[[M]]δ
ev∗γ ∈ Q[Alb(X)],
which is an element of the group algebra Q[Alb(X)] with support on the torsor T δ(L, β). Distinct
correlators may yield different Gromov–Witten invariants, as illustrated by the computations in
the elliptic case.
It follows from the discussion above that the evaluation map truly has values in the set
a−1
w (φβ(L)) =
⊔
a−1
w/δ(θ). A broader generalization of GW invariants would be obtained by allow-
ing the pullback of cohomology classes from a−1
w (φβ(L)) rather than X
n. Correlated invariants
are a particular case of these invariants where we pull-back the classes 1θ ∈ H0
(
a−1
w (φβ(L)),Q
)
,
corresponding to the components a−1
w/δ(θ) of a
−1
w (φβ(L)).
1.4 Properties of the correlated classes
1.4.1 Deformation invariance
We prove that the correlated class is deformation invariant, so that we are right to speak about
a refinement of the GW invariants. Here we are allowed to deform both the variety X and the
line bundle L, considering a family X → S with a line bundle L → X. Notice the correlators
take values in the torsor inside the Albanese variety AlbX/S which is also deformed. Using the
latter for some suitable choice of family, we furthermore prove the following.
Theorem (Theorem 3.11). The correlated invariants ⟨⟨γ⟩⟩δg,β,w are invariant under the action
of the subgroup φβ
(
Torδ
(
Pic0(X)
))
.
In the case where X = E is an elliptic curve, we have Pic0(E) ≃ Alb(E) ≃ E. If the chosen
homology class is β = a[E], the map φβ : E → E is actually the multiplication by a, so that we
have φa[E](Torδ(E)) = Torδ/ gcd(a,δ)(E). In the two extremal case, it means the following:
� If a and δ are coprime, the subgroup is Torδ(E), so that the classes
[
Mθ
]vir
all provide the
same invariants.
� If δ|a, the subgroup is trivial and we have no further information on the correlated invari-
ants.
Specific to the elliptic case, choosing L = O so that T δ(L, β) = Torδ(E), we furthermore prove
that the invariant for the class
[
Mθ
]vir
only depends on θ through its order in Torδ(E).
4 T. Blomme and F. Carocci
1.4.2 Degeneration formula
One of the most important tools in the computation of Gromov–Witten invariants is the degen-
eration formula [25, 27, 32]. Our second contribution is a degeneration formula suited for the
correlated classes.
We state two degeneration formulas. The first one is stated in Theorem 4.7 and valid
for any variety X. The drawback of the latter is that to be useful, one needs an expres-
sion for a refinement of a diagonal class. In the usual degeneration formula, such expression
is provided by the Künneth decomposition formula. In our situation, such a decomposition
is not obvious to get. The second formula deals with the particular case where the maps
H•(Xn,Q) → H•(a−1
w/δ(θ),Q
)
are surjective. Under this assumption, we recover an expres-
sion using the usual diagonal with its Künneth decomposition. To state the formula, we use the
operator d
[
1
δ
]
on Q[Alb(X)], which maps (θ) to the average of its δ-roots.
Theorem (Corollary 4.12). For any class γ ∈ H•(Xn × Y m,Q), we have the following equality
between intersection numbers:
ev∗γ ∩ [[M0]]
δ =
∑
Γ
∏
we
|Aut(Γ)|
(
ev∗γ ∪ ev∗D∨) ∩ d
[
1
δ/δΓ
]∏
V
[[MV ]]
δΓ ,
where the sum is over degeneration graphs (Definition 4.1), D∨ is the Poincaré dual of the class
of the diagonal, and d is the division operator in the group algebra.
This technical surjectivity assumption is satisfied in the case where X is an elliptic curve E
since in that case the map a−1
w/δ(θ) ↪→ Xn is just the inclusion of a subtorus.
1.5 The elliptic case
The remainder of the paper focuses on the easiest possible case, where X is an elliptic curve.
For simplicity in the introduction, we state the theorems for the case L ∼= O.
1.5.1 Local invariants
In Section 5, we are able to provide an explicit computation of the following correlated Gromov–
Witten invariants, called local GW invariants
⟨⟨pt0, 1w1 , ptw2
, . . . ,ptwn
⟩⟩δ1,a[E],w.
For this specific choice of insertions, the invariant is a concrete count of curves which are in this
situation covers of E. The enumeration is thus possible through a careful study of these covers.
Theorem (Theorem 5.3). The full local correlated invariant has the following expression:
⟨⟨pt0, 1w1 , ptw2
, . . . ,ptwn
⟩⟩δ1,a[E],w = an−1w2
1σδ(a),
where σδ(a) is a refinement of the sum of divisors function σ(a) =
∑
d|a d (see Section 5).
We provide an explicit expression for σδ(a). These invariants are in a sense the most simple
ones because they carry a unique interior point constraint. They enable the computation of
invariants with more point constraints through the degeneration formula as seen in the next
section.
Correlated Gromov–Witten Invariants 5
1.5.2 Floor diagrams and regularity
Using the degeneration formula, the above local invariants are in fact the ground blocks enabling
the computations of the more general correlated invariants ⟨⟨ptn+g−1, 1w1 , . . . , 1wn⟩⟩δg,a[E],w. The
degeneration formula reduces the computation to the enumeration of floor diagrams with some
new multiplicities.
Floor diagrams are a combinatorial tool introduced by Brugallé and Mikhalkin to enumerate
some planar curves subject to point conditions chosen in a stretched configuration [12, 13].
Many variations have since been introduced to deal with various situations, see [7] for the
version relevant for this paper.
Using the floor diagram algorithm, we are able to provide two regularity statements refining [7,
Theorems 6.6 and 6.9]. The first statement deals with the generating series in the class direction.
Theorem (Theorem 6.12). Let w be a tangency profile of length n. The generating series
∞∑
a=1
〈〈
ptn+g−1, 1w1 , . . . , 1wn
〉〉δ
g,a[E],w
qa
is quasi-modular for the congruence group Γ0(δ) with values in the group algebra C[Torδ(E)].
Compared to [7, Theorem 6.9], the refinement loses regularity since quasi-modularity is only
satisfied for the congruence subgroup Γ0(δ) and not SL2(Z). The next regularity statement
concerns the dependance in the tangency orders w. Namely, we consider the following functions:
N δ
g,a : w 7−→
〈〈
ptn+g−1, 1w1 , . . . , 1wn
〉〉δ
g,a[E],w
∈ Q[E],
defined on the set of tangency profile w divisible by δ.
Theorem (Theorem 6.16). For fixed a, g, n, δ, the function w 7→ N δ
g,a(w) is piecewise polyno-
mial.
Theorem 6.16 states that the piecewise polynomiality from [7, Theorem 6.6] survives the
refinement. Such behavior has also been observed in a number of similar situations, such as
double Hurwitz numbers and relative GW invariants of Hirzebruch surfaces [3].
1.6 Future directions
Correlated GW invariants from the elliptic case are interesting due to their relation to the GW
invariants of bielliptic surfaces, when caring about the torsion part in the homology group of
a bielliptic surface. The GW invariants up to torsion have already be computed by the first
author in [8], and the use of correlated invariants is necessary for the torsion part.
Importantly, these invariants seem also to be necessary for the study of reduced GW invariants
of abelian surfaces, and the mysterious multiple cover formula they should satisfy [6, 40]. We plan
to investigate this further in subsequent works. Indeed, looking at the reduced decomposition
formula from [31, Section 4.6], we notice the following. The hyperplane of values of the evaluation
map is in fact disconnected when tangency orders along the gluing divisor have a common non-
trivial divisor. The correlators give a way to concretely describe the connected components,
only one of them containing the diagonal involved in the decomposition formula.
Many problems concerning these correlated invariants and their computation remain open,
such as their computation in the presence of ψ-classes or λ-classes. It would also be interesting to
see if we can have a correlated double-ramification cycle formula refining the formula from [21].
6 T. Blomme and F. Carocci
2 Logarithmic curves, line bundles and stable maps
2.1 Curves and line bundles
We refer the reader to [41] for an extensive introduction to logarithmic structures and in par-
ticular for all the basic definitions which we do not recall in what follows.
2.1.1 Logarithmic curves
Let (S,MS) be a fine and saturated logarithmic scheme. A log curve over S is a proper, integral
and logarithmically smooth morphism π : C → S with connected, reduced one-dimensional
(geometric) fibers. Kato provided the following local characterization.
Theorem 2.1 ([23]). For every geometric point p ∈ C with π(p) = s, there exists an étale local
neighbourhood of p in C with a strict étale morphism to
smooth point: A1
S with log structure pulled back form the base, i.e., MC,p =MS,s;
marking: A1
S with log structure generated by the 0S-section and π∗MS, i.e., MC,p =MS,s⊕N;
node: Spec(OS [x, y]/xy − t) for some t ∈ OS, with log structure induced by the multiplication
map A2
S → A1
S and t : S → A1. In particular, MC,p = MS,s ⊕ Nex ⊕ Ney/ex + ey = δ
for δ ∈MS,s the so called smoothing parameter.
2.1.2 Tropical curves
To a family of logarithmic curves, we can associate a family of tropical curves. Consider first
C → (Spec(k), Q) a logarithmic curve over a log point.
The tropicalization Γ is defined as the generalised cone complex obtained as the following
colimit:
lim−→
η⇝ξ
Hom
(
MC,p,R⩾0
)
,
where η ⇝ ξ denote specialization maps inducing surjective morphisms MC,ξ → MC,η. From
the local description of the log structure on log smooth curves recalled above, it follows that Γ
comes equipped with a morphism of cone complexes Γ → σQ, for σQ = Hom(Q,R⩾0) the cone
dual to the monoid Q.
Alternatively, Γ can be thought as the dual graph of C (having a vertex for each irreducible
component, a leg for each marking and an edge e ∈ Γ for each node) together with a metriza-
tion of the bounded edges with values in the base monoid Q. This means that we have an
assignment l : Eb(Γ) → Q defined by associating to each bounded edge its smoothing parame-
ter δe ∈ Q.
Remark 2.2. Notice that for each map Q → R⩾0 we obtain a tropical curve in the classical
sense. Since Q → R⩾0 correspond to a point of the cone σQ dual to Q (in the usual sense of
toric geometry), we can think of a tropical curve metrized in Q as a family of usual tropical
curves on the dual cone.
Remark 2.3. The tropicalization makes sense over more general base log schemes (see also [38,
Definition 2.3.3.3]). Let (C,MC) → (S,MS) a log smooth curve; for each s ∈ S, Γs consists of
the dual graph of Cs metrized in MS,s. For each specialization η ⇝ s, we have a compatible
diagram
Γs Γη
MS,s MS,η,
c
ℓ ℓ
Correlated Gromov–Witten Invariants 7
where MS,s → MS,η is a localization morphism followed by sharpification (quotient the invert-
ibles) and c is an edge contraction; more precisely, c contracts the edges of Γs whose length
becomes zero in MS,η.
2.1.3 Piecewise linear functions
Unravelling the definition of global sections of the characteristic sheaf, we get a convenient
description of H0
(
C,M
gp
C
)
in terms of continuous functions on the tropicalization Γ which are
linear with integral slope along the edges. Let us first assume that C is a log curve over
a S = (Spec(k), Q).
Since M
gp
C is a constructible sheaf, a section is determined by giving mx ∈M
gp
C,x at a generic
point x of each stratum, such that the values mx are compatible with specialization, i.e., if ξ
generalize to η then the image of mξ under the surjection M
gp
C,ξ →M
gp
C,η is mη.
Looking at Kato’s classification, there are three type of strata on log smooth curves: the open
strata corresponding to irreducible components of the curves excluding markings, i.e., vertices in
the tropicalization; the closed strata corresponding to markings; the closed strata corresponding
to nodes. Therefore, a section α ∈ H0
(
C,M
gp
C
)
is the data of
� for each vertex V of Γ a value α(V ) ∈ Q;
� for each leg and edge an integer we ∈ Z (the slope along edge or leg in Γ);
� a compatibility condition for the values α(V ), α(W ) of vertices incident to the same edge e:
α(V )− α(W ) = δewe.
Indeed, suppose that xe is a node of C which generalizes to the generic points ηV and ηW ; then
we have a map M
gp
C,xe
→ Q×Q that is injective and gives an identification
M
gp
C,xe
= {(a, b) ∈ Q×Q | a− b ∈ δeZ}.
We denote by CL(Γ, Q) the set of functions satisfying the above conditions, called cone-wise
linear.1 The previous discussion can be summarized in the following equality (see also [30,
Section 3.3.3] and references therein) H0
(
C,M
gp
C
)
= CL(Γ, Q).
Remark 2.4. Using the description of tropicalization for a family (C,MC)
π−→ (S,MS) of log
curves given in Remark 2.3 and the above discussion, we can think of a section α ∈ H0
(
S, π∗M
gp
C
)
as a collection αs ∈ CL
(
Γs,M
gp
S,s
)
compatible under edge contraction, i.e., generalization.
2.1.4 Line bundles
Let C
π−→ S be a log smooth curve. We have a short exact sequence
0 → O×
C →Mgp
C →M
gp
C → 0, (2.1)
from which we obtain a long exact sequence of sheaves on S
· · · → π∗M
gp
C → π∗M
gp
C → R1π∗O
×
C → R1π∗M
gp
C → R1π∗M
gp
C → · · · .
In particular, if the underlying scheme of S is a point, so that π∗M
gp
C = H0
(
C,M
gp
C
)
and
R1π∗O
×
C = H1
(
C,O×
C
)
= Pic(C), we get a map H0
(
C,M
gp
C
)
→ Pic0(C), α 7→ OC(−α).
1We reserve the name piece-wise linear for those function which become linear along the edges after a subdi-
vision.
8 T. Blomme and F. Carocci
Remark 2.5. The sign comes from observing that when α ∈ H0
(
C,MC
)
, the O×
C-torsors of
lifts of α to a section of MC comes equipped with a map to OC , coming from the structure
morphism ϵ : MC → OC of the log structure. Up to changing α by −α, we now deal with OC(α)
instead.
The restriction of the line bundle OC(α) to the irreducible components CV of C can be
explicitly described [30, Proposition 3.3.3].
Proposition 2.6. Let C
π−→ S = (Spec(k), Q) be a log curve and α ∈ CL(Γ, Q), where Γ denotes
the tropicalization of C. Then for any vertex V ,
OC(α)|CV
= π∗OS(α(V ))⊗ OCV
(∑
e⊢V
se(α)qe
)
,
where the sum is taken over all the edges and legs e incident to V and se(α) is the slope of α
along e, oriented away from V , where qe denotes the preimage of the node associated to the
edge e on the considered component.
Remark 2.7. The result is more generally true if C → S is a logarithmic curve with constant
degeneration.
2.2 Logarithmic line bundles and trivializations
Definition 2.8 ([38]). Let C → S be a log smooth curve. A logarithmic line bundle P on C is
a Mgp
C -torsor such that its restriction P |Cs to the geometric fibers has bounded monodromy.
We refer the reader to [38] for all explanations about the meaning and the necessity of the
bounded monodromy. Here we only need that it is a condition on the M
gp
C -torsor P associated
to P , and that it is automatically satisfied when P is isomorphic to the trivial M
gp
C -torsor.
In particular, we see from the long exact sequence
H0
(
C,M
gp
C
)
→ Pic(C) → H1
(
C,Mgp
C
)
→ H1
(
C,M
gp
C
)
→ · · ·
that for any L ∈ Pic(C) the associated Mgp
C -torsor L× ⊗O×
C
Mgp
C is in fact a logarithmic line
bundle, where we denote by L× the Gm-torsor obtained from the line bundle removing its zero
section.
Definition 2.9 ([30, Definition 4.5.1]). Let C
π−→ S be a log smooth curve and L a line bundle
on C. A logarithmic trivialization of L is a section of L× ⊗O×
C
Mgp
C .
The definition is a modification of [30, Definition 4.5.1], where the authors define logarithmic
trivializations relative to the base. The difference comes from the fact that [30] are interested
in moduli of logarithmic maps to rubber targets, while we study the rigid case. Similarly, the
following discussion is parallel to [30, Proposition 4.5.3].
From (2.1), we see that the logarithmic line bundle L× ⊗O×
C
Mgp
C can be thought as a O×
C-
torsor over the trivial M
gp
C -torsor. In particular, there is an exact sequence
0 → H0
(
C,L× ⊗O×
C
Mgp
C
)
→ H0
(
C,M
gp
C
)
→ Pic(C),
where the second map is defined by α 7→ L(−α) = L ⊗ O(−α). In other words, a logarithmic
trivialization of L is a trivialization of the line bundle L(−α) for some cone-wise linear function α.
Correlated Gromov–Witten Invariants 9
2.3 Logarithmic stable maps to P1-bundles and their degenerations
This subsection follows closely [30, Section 5], with two small differences. First, [30] considers
maps to rubber target, while we deal with the rigid case. Second, they study maps to P1 or
to degeneration of P1 to a chain, while we consider maps to P1-bundles Y = PX(OX ⊕ L)
and degenerations of the latter to certain chains Y0 =
(
Y1, D
±
1
)
∪
(
Y2, D
±
2
)
∪ · · · ∪
(
YN , D
±
N
)
of P1-bundles over X obtained by gluing Yi to Yi+1 along D+
i
∼= D−
i+1
∼= X.
For the benefit of the reader, we state the results in the form most convenient for us and
sketch how to adapt the proofs of [30, Section 5] to our situation.
Let X be a smooth projective variety and L a line bundle on X. For a fixed β ∈ H2(X,Z)
and a vector of integers w = (w1, . . . , wn) such that
∑n
i=1wi = c1(L) ·β, we consider the moduli
space M := Mg,m
(
Y |D±, β,w
)
of logarithmic stable maps to Y = PX(OX ⊕ L) endowed with
the divisorial log structure D+ +D− = PX(OX) + PX(L) and with prescribed contact order wi
at qi along D
+ + D− [1, 19]. Positive wi encode the tangencies with D+ and the negative wi
the tangencies with D−.
This space compactifies the moduli space parametrizing maps from marked curves to X,
f :
(
C, p1, . . . , pm, q1, . . . , qn
)
→ X,
together with an isomorphism
λ : f∗L
(
−
∑
aiqi
)
∼= OC .
Using the language of logarithmic trivializations introduced in the previous subsection, this
description can be extended over the boundary.
Definition 2.10. We define M(X,L) to be the stack over LogSchfs whose objects are
(1) A log smooth curve C → S with n marked points with logarithmic structure and m
schematic marked points.
(2) A stable map f : (C, pi, qi) → X.
(3) A logarithmic trivialization λ ∈ H0
(
C, f∗L× ⊗O×
C
Mgp
C
)
, such that the induced α ∈ H0
(
C,
M
gp
C
)
is locally on C comparable with 0 and the slope of α along the i-th marking is wi.
We recall that a section α ∈ H0
(
C,M
gp
C
)
is said comparable to 0 if at each point x ∈ C we
have that either α(x) ⩾ 0 or α(x) ⩽ 0 in the partial order onM
gp
C,x defined by the monoidMC,x.
We refer the reader to [43, Section 2.5] for further explanations on the geometric meaning of
this condition.
Proposition 2.11. The moduli space M(X,L) is equivalent as a stack over logarithmic schemes
to the moduli space of logarithmic stable maps M of Gross–Siebert, Abramovich–Chen with
target (Y = PX(OX ⊕ L),MY ) and fixed tangencies wi along the markings q1, . . . , qn [1, 19].
Here MY is the divisorial log structure defined by the fiberwise 0 and ∞ sections.
Proof. The proof essentially consists in unravelling the definition of logarithmic stable map to
a P1-bundle. For the case of log stable maps to toric varieties, the analogous result is proved,
for example, in [43, Proposition 2.5.1.1].
Let Tot(L)
π−→ X be the total space of L endowed with the zero section logarithmic struc-
tureML. First, we argue that a logarithmic stable map C
F−→ Tot(L) is the same as a logarithmic
trivialization λ such that the associated α is a section of the log structure, i.e., α ∈ H0
(
C,MC
)
.
Indeed, by definition, a log map is a map F of the underlying schemes, namely f : C → X plus
a section of f∗L, together with a morphism F−1ML →MC of the logarithmic structures. Using
10 T. Blomme and F. Carocci
the Borne–Vistoli description of fine and saturated log structures [9], the latter is the same as
a morphism of the characteristic sheaves F−1ML →MC such that the maps to DivC commute.
The first map comes from ML
∼= NX → DivTot(L), 1 → OL(X) ∼= π∗L pulling back along F . It
follows that a lift of the morphism of characteristic sheaves to a morphism of the logarithmic
structures correspond to an isomorphism f∗L ∼=λ OC(α) for α the section of MC induced by the
map on characteristic sheaves.
At this point we argue parallel to [43, Section 2.5.1].
The P1-bundle PX(OX ⊕ L) with its fiberwise toric log structure can be constructed as the
quotient of the rank 2 vector bundle OX ⊕ L minus the 0 section for the fiberwise Gm-action.
Then any logarithmic map C → PX(OX ⊕ L) locally lifts to OX ⊕ L \ 0X .
By the discussion above, this lift can be represented by a, b logarithmic trivializations of OC
and f∗L such that the associatedM
gp
C section a and b are inMC . Notice that the ratio λ = ab−1
is a well defined Gm-invariant section of the torsor f∗L× ⊗O×
C
Mgp
C , namely a logarithmic trivi-
alization of f∗L.
Notice however that not all the section α ∈ H0
(
C,M
gp
C
)
can arise this way: indeed, since (a, b)
must avoid the zero section, this means that locally around each point x ∈ C either ax ∈MC,x
or bx ∈ MC,x is zero (i.e., lifts to an isomorphism of trivial Gm-torsors). For α = a − b, this
translates precisely to being locally comparable to zero. ■
Remark 2.12. If in Definition 2.10, we drop the request on the local comparability with zero
of the section α ∈ H0
(
C,M
gp
C
)
, we would obtain the stack over LogSchfs parametrizing log
maps to the Glog bundle Tot(L) ×Gm Glog associated with L. Similar moduli stack have been
considered various times in the literature [30, 42, 43, 44].2
Remark 2.13. The definition and the characterization above can easily be adapted to the case
of moduli of logarithmic stable maps to expansions [24, 26, 32]. Indeed, it is sufficient to add
the condition that the values of α(V ) are totally ordered; similar discussions appear in [30, 42].
In the classical language of [26], maps in the boundary of M which differ by the rubber action
are identified. It is explained in detail in [14] how to see this from the logarithmic prospective.
We are also interested in considering log maps to degenerations, i.e.,
� Logarithmic maps to Yt
p−→ A1
t certain one parameter degenerations of PXt ̸=0
(O⊕ L) to
Y0 =
(
Y1, D
±
1
)
∪
(
Y2, D
±
2
)
∪ · · · ∪
(
YN , D
±
N
)
,
where
(
Yi, D
±
i
)
are P1-bundles over X0 glued along their infinity and zero sections; A1
t
is endowed with its toric log structure and Yt with the divisorial log structure defined
by Y0 +D+ +D−, where D± are the 0 and ∞-section of the family, and making p log
smooth.
� Logarithmic maps to the log smooth scheme Y0 → (Spec k,N) with the log structure pulled
back from the family Yt
We recall the following definition.
Definition 2.14 ([30, Definition 5.2.1]). A divided tropical line over a logarithmic scheme
(S,MS) is defined as follows. Let P be an Mgp
S -torsor and P the associated M
gp
S torsors; fix
γ0 ⩽ γ1 ⩽ · · · ⩽ γN a non empty collection of sections of P defined locally on S. Then P γ ⊂ P
is defined to be the subfunctor whose local sections are comparable with all the γi.
2We thank an anonymous referee whose comments suggested us to add this remark.
Correlated Gromov–Witten Invariants 11
It is proved in [30, Proposition 5.2.4] that Pγ := P γ ×P P is a 2-marked family of semi-stable
genus zero curve.
We then consider the following variation: fix (X,MX)
p−→ (S,MS) with X → S smooth
projective and MX = p∗MS . The two cases of interest listed above correspond to the case
where (S,MS) is A1 with its toric log structure or a standard log point.
Let P ∈ LogPicX/S and P the induced M
gp
X -torsor. Let γ0 ⩽ γ1 ⩽ · · · ⩽ γN be a non empty
collection of sections of P . Notice that where the logarithmic structure of X is trivial P is a line
bundle in the usual sense and all the γi coincide and are necessarily zero.
Lemma 2.15. In the notation above, let P γ ⊆ P be the sub-funtcor of sections locally comparable
with all the γi and Pγ = P ×P P γ. Then Pγ → X → S is an expanded degeneration of Y =
PX(L⊕ O) for L the line bundle on X of lifts of γ0 to a section of P .
Proof. Since we assumed that the collection of sections is not empty, P admits a trivializa-
tion γ0 : X → P which we think as the zero of MX group. Notice that if the log structure on S
and thus on X is generically trivial, then all the γi coincide and are actually zero on the locus
of X with trivial logarithmic structure. Then there exist a line bundle L0 representing the log
line bundle P , namely the O×
X -torsor of lifts of γ0 to a section of the log line bundle. We al-
ready proved in Proposition 2.11 that the subfunctor of P = L×
0 ⊗O×
X
Mgp
X whose local sections
are comparable with γ0 is the P1-bundle Y = PX(L0⊕O) endowed with its boundary logarithmic
structure. Once a trivialization is fixed, we have that
γi − γ0 = δi ∈ H0
(
M
gp
X
)
;
these are maps Trop(X) = Trop(S)
δi−→ Trop(Y). Subdividing along the image, we obtained the
desired degeneration.
Alternatively, the statement can be proved following the steps of [30, Proposition 5.2.4]. ■
Let Yt → A1 a semi-stable degeneration coming form a subdivided tropical line; for example,
this is the case if Yt is constructed starting from PXt(OXt ⊕ Lt) by successively blowing up the
zero and the infinity section on the central fiber. Denote by M
(
Yt/A1
)
the moduli space of
logarithmic stable maps to Yt/A1 from families of log smooth curves with fixed genus g, number
of markings m, contact order (w1, . . . , wn) along D−
t +D+
t and curve class β. Then we have the
following.
Corollary 2.16. The moduli space M
(
Yt/A1
)
parametrizes the following data:
(1) A diagram of logarithmic maps
C Xt
S A1
f
for C → S a family of log smooth curves.
(2) A section λ ∈ H0
(
C, f∗L× ⊗O×
C
Mgp
C
)
such that the induces α ∈ H0
(
C,M
gp
C
)
is locally
comparable with f∗γi.
Remark 2.17. Parallel to before, if we want to instead consider logarithmic maps to expanded
degenerations [24, 26, 32], it suffices to further impose that for each geometric point s the values
of α(V ) for V ∈ Γs are totally ordered.
12 T. Blomme and F. Carocci
3 Correlated virtual class
3.1 Recollection on Albanese varieties
3.1.1 Complex setting
For X a proper, smooth variety over the complex numbers the Albanese variety was origi-
nally defined via transcendental methods using path integrals of closed holomorphic 1-forms.
Let H1(X,Z) denote the torsion free part of the first homology group. There is an inclusion via
the integration on paths:
H1(X,Z) −→ H0(X,ΩX)∗, γ 7−→
(
ω 7→
∫
γ
ω
)
,
where H0(X,ΩX)∗ denotes the dual to the vector space of holomorphic 1-forms. The Albanese
variety of X is the complex abelian variety obtained as the following quotient:
Alb(X) := H0(X,ΩX)∗/H1(X,Z).
We list below some results and properties of the Albanese variety and refer the reader to [5,
Section 11.11] or [39] and references therein for all details and proofs.
(1) When X is an Abelian variety, then Alb(X) ∼= X.
(2) Let x0 ∈ X a point, then there is an algebraic map, called the Albanese map
aX : X −→ Alb(X), x 7−→
(
ω 7→
∫ x
x0
ω
)
mod H1(X,Z),
sending x0 to the identity element. This map is unique up to translation in the sense that
choosing a different x0 amounts to compose aX with a translation.
(3) The Albanese map satisfies the following universal property: let φ : X → A be a map
to an abelian variety, then there exists a unique homomorphism of abelian varieties
φ̃ : Alb(X) → A such that φ̃(0) = φ(x0) and φ̃ ◦ aX = φ.
(4) Let X
f−→ Y a morphism of algebraic varieties, then the map to the Albanese varieties are
functorial, i.e., there exists a homomorphism of abelian varieties f∗ making the following
diagram commute:
X Y
Alb(X) Alb(Y ),
f
aX aY
f∗
where aY maps f(x0) to the identity.
(5) The Albanese variety Alb(X) is the dual abelian variety of the Picard group
Pic0(X) := H1(X,OX)/H1(X,Z),
i.e., Alb(X) = Pic0
(
Pic0(X)
)
. Identifying Pic0(X) with the connected component of the
identity of the group of line bundles, the functorial homomorphisms of Albanese variety f∗
is the dual of the pullback map.
Remark 3.1. Notice that in the complex setting, H1(X,OX) can be seen as sheaf cohomology
or alternatively as the Dolbeault cohomology group of (0, 1)-forms.
Correlated Gromov–Witten Invariants 13
3.1.2 Algebraic definition
The interpretation of the Albanese variety as the dual abelian variety of Pic0X/S allows a more
algebraic definition. Indeed, when X
g−→ S be a smooth, projective, morphism with connected
geometric fibers over a scheme S, Grothendieck proved that there exists a smooth S-scheme, lo-
cally of finite presentation, PicX/S representing the relative Picard functor and an abelian scheme
(in particular, smooth and proper) Pic0X/S over S whose fibers Pic0Xs/k(s)
are the connected com-
ponents of the identity in PicXs/k(s). See, for example, [17, Section 9], or [10, Chapter 8] and
references therein. Then by [20, Theorem 3.3] (see also [35, 36]), we can consider the dual
abelian scheme
Alb0X/S := Pic0
Pic0
X/S /S
=
(
Pic0X/S
)∨
.
If g : X → S has a section σS : S → X, then we get an Albanese map to Alb0X/S constructed in
the following way.
Let P be the unique Poincaré line bundle on X × Pic0X/S which trivializes along σS in the
sense of [10, Section 4].
Then we define the Albanese morphism
aX/S : X −→ Alb0X/S ,
(
U
f−→ X
)
7−→ (f × id)∗P.
Given X
f−→ Y a morphism of smooth, proper, geometrically connected S-varieties and σS : S
→ X a section, it follows from the universal property of the Albanese scheme that there exists
a unique homomorphism f∗ of S-abelian schemes making the following diagram commute:
X Y
Alb0X/S Alb0Y/S ,
f
aX/S aY/S
f∗
where aX/S and aY/S sends the sections σS : S → X respectively f ◦ σS : S → Y to the identity
element. Alternatively, f∗ can be described as the dual (in the category of abelian S-schemes,
see [35, Section I.8]) of the pullback f∗ : Pic0Y/S → Pic0X/S .
3.1.3 Self-duality for smooth curves
As explained, for example, in [17, Section 9.5.26], when C → S is a family of smooth, projective,
geometrically connected genus g > 0 curves over a Noetherian base, there exists an self-duality
isomorphism Alb0C/S → Pic0C/S .
If we have a section σ0 : S → C, then the self-duality has a very explicit description using the
Abel–Jacobi map Aσ0 : C → Pic0C/S given by
Aσ0 : C −→ Pic0C/S , (T → C) 7−→ OCT
(pT − σ0|T ),
where pT : T → C ×S T is the induced section of CT = C ×S T . The self-duality isomorphism
is simply given by the pullback along the Abel–Jacobi map. The isomorphism does not depend
on the choice, nor on the existence of the section, see, for example, [16] for a proof.
Notice that Aσ0 coincides with the Albanese map defined using the Poincaré line bundle
on X× Pic0X/S rigidified along the section σ0.
3.2 Refinement of the moduli spaces
Let M = Mg,m
(
Y |D±, β,w
)
be the moduli space of logarithmic stable maps to Y = P1
X(O⊕L)
considered in the previous section and let us now assume that L ∈ Pic0(X): we have c1(L) = 0.
14 T. Blomme and F. Carocci
3.2.1 Morphism to Alb(X)
Using the tangency orders w, we consider the map
aw : (xi) ∈ Xn 7−→
n∑
i=1
wiaX(xi) ∈ Alb(X).
As
∑
wi = 0, this map does not depend on the choice of a base point in X and is uniquely
defined. Indeed, for z0, z
′
0 two choices of base point and aX , a′X the corresponding morphisms,
we have that a′X = ta′X(z0) ◦ aX and thus
n∑
i=1
wia
′
X(xi) =
n∑
i=1
wi(aX(xi)− a′X(z0)) =
n∑
i=1
wiaX(xi).
Now, consider the morphism from the moduli space to the Albanese variety of X by composing
with the evaluation map:
κ : M
ev−→ Xn aw−−→ Alb(X)
defined by(
C
f−→ X, (pj)
m
j=1, (qi)
n
i=1,OC(α) ∼=λ f∗L
)
7→ (xi := f ◦ qi)ni=1 7→
n∑
i=1
wiaX(xi).
Recall that (qi)
n
i=1 are marking with logarithmic structure while (pj)
m
j=1 are standard schemat-
ical points on C.
Lemma 3.2. Let β ∈ H2(X,Z) be the homology class of a complex curve. The bilinear map
ϕ : α⊗ ω ∈ H0,1(X)⊗H1,0(X) 7−→
∫
β
α ∧ ω ∈ C
induces a morphism
φβ : Pic0(X) −→ Alb(X).
Proof. We use the isomorphisms H0,1(X) = H1(X,OX) and H1,0(X) = H0(X,ΩX) as well
as the Hodge decomposition H2(X,C) = H0,1(X) ⊕ H1,0(X). Through the above isomor-
phisms, the map H1(X,Z) → H1(X,OX) yielding Pic0(X) is actually the composition of
H1(X,Z) → H1(X,C) followed by the projection onto H0,1(X). The bilinear map ϕ induces
a linear map ϕ∗ : H0,1(X) → H1,0(X)∗. To prove that it descends to a map Pic0(X) → Alb(X),
we need to show that
ϕ∗
(
H1(X,Z)
)
⊂ H1(X,Z) ⊂ H0(X,ΩX)∗.
Let ω ∈ H1,0(X) and γ ∈ H1(X,Z). We write γ ⊗ C = γ0,1 + γ1,0 for its Hodge decomposition
in H2(X,C). We have
ϕ∗(γ)(ω) =
∫
β
γ0,1 ∧ ω (by definition)
=
∫
β
(
γ0,1 + γ1,0
)
∧ ω = β ∩ (γ ∪ ω) = (β ∩ γ) ∩ ω,
where the second equality follows from the fact that
∫
β γ
1,0 ∧ ω = 0, since β is the class of
a complex curve. The last line ensures that ϕ∗(γ)(ω) is a period of ω since β ∩ γ ∈ H1(X,Z).
Therefore, as desired, H1(X,OX) → H0(X,ΩX)∗ descends to a map
φβ : H1(X,OX)/H1(X,Z) −→ H0(X,ΩX)∗/H1(X,Z). ■
Correlated Gromov–Witten Invariants 15
Remark 3.3. Giving a non-degenerate positive bilinear map H1(X,OX) ⊗ H0(X,ΩX) → C
amounts to providing a polarization, i.e., an ample line bundle L, on the Abelian variety Pic0(X),
see, for example, [39]. In an Abelian variety, a polarization then induces a map
AL : L ∈ Pic0(X) 7−→ t∗LL⊗ L−1 ∈ Pic0
(
Pic0(X)
)
= Alb(X).
Assume X is projective with hyperplane class h ∈ H2(X,Z) giving a Kähler form on X. Then
we have a polarization given by the non-degenerate positive bilinear form
ϕ : α⊗ ω 7−→
∫
X
hn−1 ∧ α ∧ ω.
Assume the class β is obtained by intersecting sufficiently many hyperplane sections: β is
Poincaré dual to hn−1. Then, the previously defined morphism φβ is actually the morphism AL
provided by the above polarization on Pic0(X).
We claim that the morphism κ is in fact constant to φβ(L). To prove the latter, we start
with the case of smooth curve, and then use it to deal with the case of nodal curves.
Proposition 3.4. The morphism κ : M → Alb0(X) defined above is constant equal to φβ(L).
Proof. Let us first look at a geometric point s = Spec(C) → M of the moduli space such that the
source curve C is smooth. We claim that in such case κ(s) = f∗f
∗L, where OC
(∑n
i=1wiqi
) ∼= f∗L
(by definition of the moduli functor) and f∗ : Pic0(C) → Alb0(X) is the morphism obtained
dualizing f∗ : Pic0(X) → Pic0(C) via the self-duality isomorphism of Pic0(C).
To see that, choose p0 a point in C and z0 = f(p0); then the functoriality of the Albanese
morphism for compatible choices of base point gives
f∗(f
∗L) = f∗
(
OC
( n∑
i=1
wiqi
))
= f∗
( n∑
i=1
wiAp0(qi)
)
=
n∑
i=1
wiaX(xi)
with the last equality following from the commutativity.
Since we are working over C, we may use that for X (resp. C), we have
Pic0(X) = H1(X,OX)/H1(X,Z) and Alb(X) = H0(X,ΩX)∗/H1(X,Z).
Through Hodge theory, H1(X,OX) is isomorphic to the Dolbeault cohomology group H0,1(X).
In particular, by functoriality, the pullback map f∗ : Pic0(X) → Pic0(C) is in fact induced
by f∗ : H1(X,OX) → H1(C,OC), which is the pullback at the level of (0, 1)-forms. Using that
H0(X,ΩX) = H1,0(X), the composition f∗f
∗ is induced by the C-linear map
φC : H0,1(X)
f∗
−→ H0,1(C) ≃ H1,0(C)∗
f∗−→ H1,0(X)∗,
where the second arrow is the dual to the pullback map for holomorphic forms. The middle
isomorphism is provided by the Poincaré duality. Finally, if α ∈ H0,1(X) is a (0, 1)-form on X
and ω ∈ H1,0(X) an holomorphic 1-form, we have
φC(α)(ω) =
∫
C
f∗α ∧ f∗ω =
∫
C
f∗(α ∧ ω) =
∫
f(C)
α ∧ ω =
∫
β
α ∧ ω,
where the penultimate equality is push-pull formula.
Now, assume that C is nodal; we write ν :
⊔
CV → C for the normalization map and
fV : CV → X for the restriction of f ◦ ν to the component CV of the normalization. Since
each CV is now smooth, the discussion above tells us that the morphism
fV ∗f
∗
V : Pic0(X) → Pic0(CV ) → Alb0(X)
16 T. Blomme and F. Carocci
is equal to φβV
, where βV = fV ∗[CV ]. Furthermore, since
∑
βV = β, we have
φβ =
∑
φβV
: Pic0(X) →
∏
V
Pic0(CV ) → Alb0(X), L→ ν∗f∗L→
∑
V
fV ∗(f
∗
V L).
To conclude the proof, we only need to argue that∑
V
fV ∗(f
∗
V L) = κ(s).
When C is singular, f∗L ∼=λ OC(α) for α ∈ H0
(
C,M
gp
C
)
such that
OC(α)|CV
= OCV
(∑
e⊢V
se,V (α)qe
)
,
satisfying the balancing condition
∑
e⊢V se,V (α) = 0. Here se,V (α) is the outgoing slope of α
at V . Notice that α induces an orientation on the edges of Γ. We denote with we the slope along
an edge with this induced orientation. Then if V , W are the two vertices adjacent to a bounded
edge e, we will have se,V (α) = −se,W (α) with one of the two being we.
We know from the analysis for the smooth case that
fV ∗(f
∗
V L) =
∑
e⊢V
se,V (α)aX(fV ◦ qi).
From the observation above and the fact that fV (qe) = fW (qe) since the maps are induced by
the maps on the nodal curves, it follows that when we consider
∑
V fV ∗(f
∗
V L) the contributions
coming from the nodes cancel out and we obtain the required identity. ■
Example 3.5. Assume X = E is an elliptic curve. We have Alb(E) ∼= Pic0(E) ∼= E. For the
homology class β = a[E], the associated map φβ is just the multiplication by a.
Remark 3.6. For a family C/S
F−→ X × S with non-smooth family of source curves, it is not
obvious how to define
F∗ : Pic
[0]
C/S → Alb0(X)× S.
Indeed, Pic
[0]
C/S → S is not an abelian scheme, as it is no longer proper, and we do not have an
Abel–Jacobi map. There are two possible ways to fix the situations.
� We can consider the compactified Jacobian Pic
[0]
C/S ⊆ JEπ ,σ
C/S parametrizing rank one, torsion
free simple (Eπ, σ)-quasi-stable sheaves on the fibers of C → S for Eπ the canonical degree 0
polarization on C/S in the sense of [33, Section 2.1]. We refer the reader to [33] and
to [34] for existence and properties of the compactifications. Then a family version of the
argument in [34, Section 5] shows that there is an isomorphism of S-group
Pic
[0]
C/S → Pic0
(
JEπ ,σ
C/S
)
and we can then define F∗ as the dual of F
∗ composed with the natural inclusion of Pic
[0]
C/S
with the compactified Jacobian.
� Alternatively, one should in future be able to use the theory of logarithmic abelian va-
rieties and of logarithmic Picard group developed by Molcho–Wise [38]. They show
that LogPicCv/S → S (where the superscript Cv means we are considering C → S en-
dowed with the vertical log structure) is a proper group stack over LogSch whose fibers
are logarithmic abelian varieties in the sense of Kajiwara, Kato and Nakayama [22]. It is
Correlated Gromov–Witten Invariants 17
expected that many classical results about abelian varieties admit generalization for loga-
rithmic abelian varieties. In this spirit, in a forthcoming paper [37] Molcho–Ulirsch–Wise
will show that LogPicCv/S is self dual in the category of logarithmic Abelian variety, ad-
mits a Abel–Jacobi map and is universal for morphism to abelian groups over log schemes.
Once such a theory will be fully developed, considering the composition
Pic
[0]
C/S ↪→ LogPicCv/S
F∗−→ Alb(X)× S
we would obtain the desired extension.
Once the extension F∗ is defined, one can look at the closed points to give an explicit de-
scription of F∗F
∗L and verify that this coincide with
∑n
i=1wiaX(f ◦ qi). Since we do not need
the extension at the level of universal Jacobian but only the map from M, we do not fill in the
technical details left out in this remark.
3.2.2 Refinement by correlation
Let us go back to the setting of M(X,L) where we fix the ramification profile. To simplify the
notation, for a given point in M, we set xi = f ◦ qi. We are interested in the case where the
contact orders have a non-trivial common divisor, i.e., there is some 1 ̸= δ| gcd(wi); the we can
consider the morphism
aw/δ : (xi) ∈ Xn 7−→
n∑
i=1
wi
δ
aX(xi).
Composing with the evaluation map, we define
κδ : M −→ T δ(L, β) ⊂ Alb(X),(
f : C → X, (pj)
m
j=1, (qi)
n
i=1,OC(α)|[f ] ∼=λ f∗L
)
7−→
n∑
i=1
wi
δ
aX(xi). (3.1)
In fact, κδ composed with the multiplication by δ is the previously defined κ. As κ is constant
equal to φβ(L) by Proposition 3.4, this ensures that κδ has values in T δ(L, β), the set of δ-
roots of φβ(L). The latter is a torsor under the finite group Torδ(Alb(X)) of δ-torsion elements
in Alb(X). In the particular case where L = OX , T δ(X,OX) is exactly the set of δ-torsion
elements in the Albanese variety Alb(X).
As L does not possess any canonical δ-root, Proposition 3.4 no longer applies for κδ, which
has no reason to be constant anymore. The morphism κδ defined in (3.1) thus determines
a decomposition of the moduli space of logarithmic maps to PX(OX ⊕ L) into components,
according to the value taken by κδ. This refinement is called correlation, as it stems from
relations between the images of the points inside the Albanese variety. The image κδ(f) is
called a correlator and is usually denoted by θ.
Remark 3.7. The preimages
(
κδ
)−1
(θ) of elements θ ∈ T δ(L, β) are called components of the
moduli space but they are not necessarily connected.
In particular, it follows from the discussion above that we have the following.
Proposition 3.8. Let M be the moduli space of logarithmic stable maps to Y = PX(O⊕L) with
the divisorial log structure defined by D− +D+. Assume that L ∈ Pic0(X) and that δ| gcd(wi).
Then we have a splitting of the virtual class[
M
]vir
=
∑
θ∈T δ(L,β)
[
Mθ
]vir
,
where T δ(L, β) is the Torδ(Alb(X))-torsor of δ-roots of φβ(L).
18 T. Blomme and F. Carocci
Definition 3.9. We encompass the information of the correlated virtual classes in the full
correlated virtual class
[[M]]δ =
∑
θ∈T δ(L,β)
[
Mθ
]vir · (θ),
which is an element in the group algebra Q[Alb(X)] with cycle coefficients and support over the
torsor T δ(L, β).
3.2.3 Correlated Gromov–Witten invariants
The projection PX(O⊕L) → X identifies D− and D+ with X. The evaluation map hence takes
the following form:
ev : Mg,m
(
Y |D±, β,w
)
−→ Y m ×Xn,
we integrate pullback of cohomology classes over the full correlated virtual class to get correlated
Gromov–Witten invariants: for γ1, . . . , γm ∈ H•(Y,Q) and γ̃1, . . . , γ̃n ∈ H•(X,Q), we set
〈〈
γ1, . . . , γm, γ̃1, . . . , γ̃n
〉〉δ
g,β,w
=
∫
[[M]]δ
m∏
1
ev∗i (γi)
n∏
1
ẽv∗i (γ̃i),
which is an element in Q[Alb(X)] with support on the torsor T δ(L, β). As the computations
in the elliptic case from Section 5 illustrate, distinct correlators may provide different Gromov–
Witten invariants, so that the refinement is non-trivial.
3.3 Deformation invariants and relations of correlated classes
Let (X,L)
p−→ B be a family of smooth projective varieties over B and let L ∈ Pic0(X/B) (in fact
log smoothness is sufficient). Let Y = P(O⊕L) → B the associated family of P1-bundles, with the
natural boundary structure D = D−+D−. Fix β an effective curve class on X relative to B and
fix contact order w = (w1, . . . , wn) of (q1, . . . , qn) with the boundary. As in the previous section,
the results of [1, 19, 32] we have that the moduli stack MB := Mg,m(Y|D, βb,w) parametrizing
diagrams of logarithmic maps
C (Y|D)
S B
F
is a Deligne–Mumford stack, proper over B end endowed with a perfect obstruction theory
Rπ∗F
∗T log
Y/B relative to B.
As for the case of absolute stable maps, for B regular and connected, the basic properties of
the virtual class construction [4, 29] ensure that the degree of the class ev∗γ ∩
[
Mb
]vir
does not
depend on b ∈ B; we also refer the reader to [28, Appendix A] and references therein for more
details on the deformation invariance.
As MB is a special case of Corollary 2.16, we still have the description of the moduli space
in terms of logarithmic trivializations. As in the case of B = Spec(C), we have the map
κB : MB → Alb(X/B)
defined by
(
C
f−→ X, (pj)
m
j=1, (qi)
n
i=1, f
∗L ∼= OC(α)
)
→
n∑
i=1
wiaX/B(f ◦ qi)
Correlated Gromov–Witten Invariants 19
which is constant with valued φβb
(Lb) on fibers of B. In particular, if δ divides gcd(wi), we get
a refinement morphism
κδB : MB −→ T δ
B(X,L) ⊂ Alb0(X/B),
where T δ
B(X,L) is a the torsor under the B-group Torδ
(
Alb0(X/B)
)
of δ-roots of φβb
(Lb)
over B. This torsor is not necessarily trivial, but we can always choose B′ → B étale such
that T δ
B(X,L)×B B
′ ∼= T δ
B′(X′,L′) trivializes.
Thus, working étale locally on B we can always assume that T δ
B(X,L) has a section over B,
i.e., it is trivial. Then MB split into connected component Mθ
B indexed by the sections θ : B →
T δ
B(X,L).
Theorem 3.10. The correlated Gromov–Witten invariants are deformation invariant, i.e.,
let X → B and MB → B as before and assume that T δ
B(X,L) has a trivializing section θ.
Then for any such section the degree of classes ev∗γ ∩
[
M
θ(b)
b
]vir
for γ ∈ H∗(Yn+m) does not
depend on b ∈ B.
Proof. Since Mθ
B → MB is open, the perfect obstruction theory Rπ∗F
∗T log
Y/B relative to B re-
stricts to Mθ
B. The results then follows once again from the propertied of the virtual class
construction [4, 29] as explained for example in [28, Appendix A]. ■
By choosing suitable families of deformations of Y → B, we can exploit Theorem 3.10 to find
non-trivial identities among the correlated classes.
The morphism φβ : Pic0(X) → Alb(X) induces a morphism between their δ-torsion el-
ements. We thus have the subgroup φβ
(
Torδ
(
Pic0(X)
))
⊂ Torδ(Alb(X)). In particular,
the support T δ(L, β) of the correlated Gromov–Witten invariants is stable by the action of
φβ
(
Torδ
(
Pic0(X)
))
. Furthermore, we have the following.
Theorem 3.11. The correlated invariants ⟨⟨γ1, . . . , γm, γ̃1, . . . , γ̃n⟩⟩δg,β,w are invariant under the
action of φβ
(
Torδ
(
Pic0(X)
))
.
Proof. We consider the family over B = Pic0(X) induced by the universal line bundle: the fiber
over L ∈ Pic0(X) is P(O⊕ L). Thus, in this case, the torsor appearing before Theorem 3.10 is
T δ
B(X,L) =
{
(L, θ) ∈ Pic0(X)×Alb(X) s.t. δθ = φβ(L) ∈ Alb(X)
}
.
To get a section, we use the base change µδ : L 7→ L⊗δ from Pic0(X) to itself. We thus have
µ∗δT
δ
B(X,L) =
{
(L, θ) ∈ Pic0(X)×Alb(X) s.t. δθ = φβ
(
L⊗δ
)
∈ Alb(X)
}
.
This torsor now has a section provided by L 7→ θ0(L) = φβ(L). Deformation invariance tells
us that for any choice of R, the correlators θ0(L) and θ0(L ⊗ R) provide the same invari-
ants. However, the latter are correlators for the a priori distinct P1-bundles associated to L⊗δ
and L⊗δ ⊗R⊗δ. Assuming R is of δ-torsion, these P1-bundles are the same. Furthermore, the
correlators differ by
θ0(L⊗R)− θ0(L) = φβ(L⊗R)− φ(L) = φβ(R),
yielding the result. ■
Finally, we have the following relations between the classes corresponding to different levels
of refinement.
20 T. Blomme and F. Carocci
Lemma 3.12. The multiplication by d in Alb(X) induces a morphism of Q[Alb(X)] denoted
by m [d]. The different full refined virtual classes are related as follows:
m
[
δ
δ′
] (
[[M]]δ
)
= [[M]]δ
′
.
Proof. The relation merely comes from the fact that multiplication by δ/δ′ provides a surjection
from δ-roots of φβ(L) to δ
′-roots of φβ(L), so that if θ′ ∈ T δ′(L, β), then we have
Mθ′ =
⊔
(δ/δ′)θ=θ′
Mθ. ■
4 Refined decomposition formula
Our goal in this section is to give a refinement of the degeneration formula [15, 27] keeping track
of our refinement.
4.1 Recollection on the degeneration formula
We consider Yt
p−→ A1 a semi-stable degeneration of a P1-bundle PXt ̸=0
(L ⊕ O) → A1
t̸=0 coming
from a subdivided tropical line as in the setting of Corollary 2.16.
As observed above, that is for example the case when Yt
p−→ A1 is constructed starting from
a family of P1-bundles over X → A1 by successively blowing up the boundary divisors on the
central fiber.
We recall some notation; write the N components of the central fiber as follows:
Y0 =
(
Y1, D
±
1
)
∪
(
Y2, D
±
2
)
∪ · · · ∪
(
YN , D
±
N
)
,
where Yi and Yi+1 are gluing along D+
i
∼= D−
i+1.
Endowing A1 with its toric log structure and Yt with the divisorial log structure coming
from Y0 +D−
1 +D+
N , p is logarithmically smooth. Let C(Y)
p−→ R⩾0 denote the tropicalization.
Then the fiber over any point of R>0, which we denote by Y, is a subdivided real line with N
vertices corresponding to the irreducible components of Y0 and two legs corresponding to the
divisors D−
1 , D
+
N .
Since Yt
p−→ A1 is logarithmically smooth, the moduli space M(Yt/A1) of (expanded) logarith-
mic stable maps to Yt/A1 admits a perfect obstruction theory relative to A1 [15, 19, 26, 32]. In
particular, there is a well defined virtual class
[
M0
]vir
for the moduli space of logarithmic stable
maps to the central fiber Y0 and moreover, as recalled above, the logarithmic invariants defined
integrating
[
M0
]vir
and
[
Mt
]vir
coincide. This allows one to compute the invariants for maps to
the smooth fiber on the degenerate fiber.
The degeneration (or decomposition) formula [15, 25, 27] expresses the virtual fundamental
class of the central fiber
[
M0
]vir
as a sum of virtual classes over a collection of decorated graphs,
which we call degeneration graphs; these encode combinatorial types of curves in the central
fiber Y0.
3
Definition 4.1. A degeneration graph for Y0 =
⋃
Yi is a graph Γ with the following decorations:
(1) We have a graph map ϕ : Γ → Y, inducing an orientation on the edges of Γ;
(2) Vertices are decorated with: a genus gV ∈ Z>0 and a curve class βV ∈ H2(Yϕ(V ),Z),
3We warn the reader that this terminology is not standard; in [27] degeneration graphs are called admissible
triples, in [25] they are referred to as bipartite decorated graph
Correlated Gromov–Witten Invariants 2122 THOMAS BLOMME, FRANCESCA CAROCCI
φ
Γ
Y
v1
v2 v3
v4
v5v5
2
2
4
4
2
1
1
2
6
Figure 1. Example of degeneration graph
Remark 4.2. In the above definition, balancing means that for every vertex V , the sum of
incoming weights matches the sum of outgoing weights. ♦
Example 4.3. In Figure 1 we draw an example of C(Y) → R⩾0 (on the left) and a corresponding
example of degeneration graph (on the right). We represent both the markings carrying
non trivial contact order, namely unbounded edges and the internal marking ( only Cv3 has
an internal marking here). Furthermore each edge, bounded or not, is decorated with the
absolute value of its weight; the sign is then determined by the natural orientation.
Finally each vertex v is further decorated with the data of the genus gv of the source curve
and a curve class βv ∈ H2(X,Z). We represent with a cross the components with gv = 0 and
βv a multiple of the fiber in Yφ(v).
♢
Let Γ be a degeneration graph. Let V be a vertex decorated with genus gV , class βV .
Let nV be the number of adjacent edges in Γ with non-zero weight and mV the number of
adjacent legs with trivial weight (called marked points). Let we = (we)e⊢V be the collection
of signed adjacent weights with sign prescribed by the divisors, so that
∑
e⊢V we = 0. We
have a moduli space of logarithmic maps associated to each vertex:
MV = MgV ,mV (Yφ(V )|Dφ(V ), βV ,we).
Let Γ̃ be a degeneration graph with a chosen labelling of its bounded edges. For a fixed
Γ there are |Eb(Γ)|!/|Aut(Γ)| such labellings. The cycle version of degeneration formula, as
presented by [Kim10] (see also [Li02, Che13] for previous versions) states that
(6) [M0]vir =
∑
Γ̃
lΓ
|Eb(Γ)|!
F∗φ
∗
Γ∆!
(∏
V
[MV ]vir
)
.
We recall the notation and refer to [KLR18] for the proof. We denoted by lΓ = lcm({we});
by ∆! the Gysin pull-back along the diagonal inclusion
D :=
∏
e
X → X =
∏
V
XnV ;
by φΓ : MΓ̃ →
⊙
V MV the étale map of degree
∏
e∈Eb(Γ) we
lΓ
to the fiber product
Figure 1. Example of degeneration graph.
(3) Every oriented edge e and every leg l (also called unbounded edges) comes decorated with
a weight we, wl ∈ Z such that the map ϕ is balanced.
(4) The flow through a vertex V , defined as the sum of the outgoing positive weights, is equal
to βV ·
[
D±
ϕ(V )
]
.
In what follows, we say edge and use the notation e for both edges and legs, unless the distinction
is important.
An automorphism of a degeneration graph Γ is an automorphism of the underlying graph
compatible with the decorations. We denote by Aut(Γ) the group of automorphisms.
Remark 4.2. In the above definition, balancing means that for every vertex V , the sum of
incoming weights matches the sum of outgoing weights.
Example 4.3. In Figure 1, we draw an example of C(Y) → R⩾0 (on the left) and a corresponding
example of degeneration graph (on the right). We represent both the markings carrying non-
trivial contact order, namely unbounded edges and the internal marking (only Cv3 has an internal
marking here). Furthermore, each edge, bounded or not, is decorated with the absolute value
of its weight; the sign is then determined by the natural orientation.
Finally, each vertex v is further decorated with the data of the genus gv of the source curve
and a curve class βv ∈ H2(X,Z). We represent with a cross the components with gv = 0 and βv
a multiple of the fiber in Yϕ(v).
Let Γ be a degeneration graph. Let V be a vertex decorated with genus gV , class βV . Let nV
be the number of adjacent edges in Γ with non-zero weight and mV the number of adjacent
legs with trivial weight (called marked points). Let we = (we)e⊢V be the collection of signed
adjacent weights with sign prescribed by the divisors, so that
∑
e⊢V we = 0. We have a moduli
space of logarithmic maps associated to each vertex MV = MgV ,mV (Yϕ(V )|Dϕ(V ), βV ,we).
Let Γ̃ be a degeneration graph with a chosen labelling of its bounded edges. For a fixed Γ,
there are |Eb(Γ)|!/|Aut(Γ)| such labellings. The cycle version of degeneration formula, as pre-
sented by [24] (see also [15, 27] for previous versions) states that[
M0
]vir
=
∑
Γ̃
lΓ
|Eb(Γ)|!
F∗ϕ
∗
Γ∆
!
(∏
V
[
MV
]vir)
. (4.1)
We recall the notation and refer to [25] for the proof. We denoted by lΓ = lcm({we}); by ∆! the
Gysin pullback along the diagonal inclusion
D :=
∏
e
X → X =
∏
V
XnV ;
22 T. Blomme and F. Carocci
by ϕΓ : MΓ̃
→
⊙
V MV the étale map of degree
∏
e∈Eb(Γ) we
lΓ
to the fiber product
⊙
V MV
∏
V MV
D X∆
corresponding to lifts of the maps in
⊙
V MV to a log stable map to Y0; by F : M
Γ̃
→ M0 the
natural clutching map, shown [24, Lemma 9.1] to have virtual degree |Eb(Γ)|!
lΓ
.
As also explained in [25], and in full details in [18, Chapter 19], via the cycle map A∗
cl−→ H•,
we can reduce to perform the computations of the invariance in (Borel–Moore) homology rather
then Chow homology. In particular, by [18, Section 19.1, Theorem 19.2]
cl
(
∆!
(∏
V
[
MV
]vir))
= cl
((∏
V
[
MV
]vir)) ∩ ev∗D∨
in H•(
∏
V
[
MV
]vir
) where D∨ ∈ H•(X) is the Poincaré dual of the class of the diagonal. Then
using push-pull formula along the following diagram:
M
Γ̃
M0
⊙
V MV
∏
V MV Y0
F
ϕΓ
ev
ev
and the identity above, we then obtain as an immediate corollary from (4.1) the following
numerical version of the degeneration formula. Using the Künneth decomposition of D∨, we can
split the cap product over the vertices and get〈
n∏
i=1
γi
〉M0
g,β
=
∑
Γ
∏
ewe
|Aut(Γ)|
∫
∏ [
MV
]vir∏ ev∗i γi ∪ ev∗D∨. (4.2)
4.2 Toward the refined degeneration formula
For Yt → A1, a degeneration coming from a divided tropical line, the interpretation of the moduli
space given in Corollary 2.16 allows us to define the refinement morphism κA1 : M
(
Yt/A1
)
→
Alb
(
X/A1
)
as the Albanese evaluation, i.e.,
(
C
f−→ X, (pj)
m
j=1, (qi)
n
i=1, f
∗L ∼= OC(α)
)
→
n∑
i=1
wiaX/A1(f ◦ qi).
The same argument given in the second part of Proposition 3.4 to compute the κ(s) in the
case of a semi-stable curve allows us to show that κ0 : M(Y0/ Spec(N → C)) → Alb(X0) is the
constant morphism φβ0(L0). Therefore, as in the case of smooth one parameter families we
have that if δ divides gcd(wi) there is a morphism κδA1 : M
(
Yt/A1
)
→ T δ
A1(X,L). Up to passing
to an étale neighbourhood of 0 ∈ A1, we can assume that T δ
A1(X,L) trivializes. In particular,
we obtain a decomposition of M0 = M(Y0/ Spec(N → C)) into connected components indexed
by θ ∈ T δ
A1(X,L)|0 = T δ(X0,L0)[
M0
]vir
=
∑
θ∈T δ(X0,L0)
[
Mθ
0
]vir
.
Correlated Gromov–Witten Invariants 23
Our goal is now to prove a decomposition formula for the components
[
Mθ
0
]vir
.
Let Γ be a degeneration graph; for each vertex V let δV is the g.c.d. of the weights of the
edges adjacent to V . Then, as discussed in Section 3.2.2, we have a refinement[
MV
]vir
=
∑
θV ∈T δV
[
M
θV
V
]vir
,
where T δV is the torsor providing the refinement for Yϕ(V ). We aim to describe the compo-
nents
[
Mθ
0
]vir
in terms of the refined components MθV
V .
Remark 4.4. Clearly several choices of θ = (θV ) may contribute to a commonMθ
t on the smooth
fiber. On the other hand, as we will see shortly, a fixed collection θ = (θV ) may contribute to
several distinct Mθ
0 and should thus not be considered as a further refinement of the Mθ
0.
For this reason, the best way to express the refined degeneration formula is in term of full
correlated virtual class (see Definition 3.9):
[[M]]δ =
∑
θ∈T δ(L,β)
[
Mθ
]vir · (θ) ∈ H•(M,Q)⊗Q[Alb(X)].
In the group algebra Q[Alb(X)], we also have the division operators defined as follows on
generators
d
[
1
d
]
(θ) =
1
d2r
∑
dθ′≡θ
(θ′),
where r is the rank of Alb(X), which is thus a real torus of dimension 2r. In other words,
d
[
1
d
]
maps an element to the average of its d-roots.
4.3 General degeneration formula
Let Γ be a degeneration graph and V one of its vertices. Let δΓ be the g.c.d. of the diagram, i.e.,
the gcd of the edge weights. Let nV be its valency, by which we mean the number of adjacent
edges and legs, and mV the number of marked points with trivial contact order. We have the
evaluation map evV : MV → XnV and the Albanese map
aV : (xe)e⊢V ∈ XnV 7−→
∑
e⊢V
we
δV
aX(xe) ∈ Alb(X).
The refinement morphism κV is the composition aV ◦ evV .
By definition, the evaluation map evV restricted to the connected component M
θV
V takes
values in H
θV
V = a−1
V (θV ).
Let θ = (θV ) be a vector of correlators indexed by vertices; we denote
Mθ =
∏
V
M
θV
V , Hθ =
∏
V
H
θV
V .
Furthermore, we denote by ẽvV the evaluation map restricted to M
θV
V (omitting θV in the
notation) and by ẽv =
∏
V ẽvV , still omitting θ in the notation.
24 T. Blomme and F. Carocci
As before, we have X =
∏
V X
nV and a natural inclusion ι : Hθ ↪→ X the inclusion, so that
we have ev = ι ◦ ẽv. All the data sits in the following diagram:
Mθ D
Hθ X Alb(X)|V(Γ)|.
Alb(X)
ẽv ev
ι
aw/δ
a
As above, D denotes the diagonal inclusion for each pair of coordinates corresponding to
a bounded edge of Γ. The components of a are the aV , while aw/δ is the morphism giving the
global refinement, i.e., aw/δ(x) =
∑
eleg
we
δ aX(xe).
Clearly, the evaluation map
∏
MV
ev−→ X may be factored via
⊔
Hθ and refined by θ⊔
θ
Mθ
⊔
θ ẽv−−−→
⊔
θ
Hθ ↪→ X.
Intersecting the diagonal D with the codomain, the latter also gets refined into
⊔
θ D
θ, where
we set Dθ = D ∩Hθ. Therefore, the fiber product
⊙
MV also gets refined into
⊔
θ(
⊙
MV )
θ,
and we have the following cartesian diagram:⊙
V MV
∼=
⊔
θ(
⊙
MV )
θ
∏
V MV =
⊔
θ M
θ.
⊔
θ D
θ ⊔
θ H
θ
D X
ẽv
∆
The key is that each Dθ is itself disconnected and the various distinct components can
contribute to distinct Mθ
0. This contribution is controlled by the following lemma.
Lemma 4.5. Let x ∈ Dθ. Then, it satisfies
δ
δΓ
aw/δ(x) =
∑
V
δV
δΓ
θV .
Proof. Let x =
(
xVe
)
∈ Dθ. We have the following:
δ
δΓ
aw/δ(x) =
δ
δΓ
∑
eleg
we
δ
aX(xe) =
∑
eleg
we
δΓ
aX(xe) =
∑
V
∑
e⊢V
we
δΓ
aX(xV,e)
=
∑
V
δV
δΓ
aV (xV ) =
∑
V
δV
δΓ
θV ,
where the third equality follows from the balancing condition that has to be satisfied by those
maps that glue to a map to Y0. ■
Remark 4.6. The proof relies on the fact that aw/δ belongs to the Q-span of the aV : we have
the identity δ
δΓ
aw/δ =
∑ δV
δΓ
aV . We may assume that δΓ|δ by choosing δ to be the g.c.d. of the
tangency orders. For other δ, we always can unrefine using Lemma 3.12.
Correlated Gromov–Witten Invariants 25
In particular, aw/δ
(
Dθ
)
can take a discrete set of values, indexed by the torsor of (δ/δΓ)-roots
of
∑ δV
δΓ
θV ; these values index disjoint components of Dθ. Denote by Dθ,θ be the component
of Dθ corresponding to θ ∈ aw/δ
(
Dθ
)
and let ∆θ,θ be the restriction of ∆ to this component
then ∆θ =
∑
θ ∆
θ,θ. If L is chosen generically, we can assume that Hθ and Dθ are manifolds
since they are defined transversally. In particular, ∆θ is a regular embedding.
Theorem 4.7. In the notation of this section, we have a refined decomposition[
Mθ
0
]vir
=
∑
Γ̃
∑
θ
lΓ
|Eb(Γ)|!
F∗ϕ
∗
Γ∆
θ,θ,!
(∏
V
[
M
θV
V
]vir)
,
where the second sum is over the θ = (θV ) such that
∑ δV
δΓ
θV = δ
δΓ
θ.
Proof. The proof follows from the usual decomposition formula and Lemma 4.5. The compat-
ibility of the Gysin pullback from [18, Chapter 6] ensures that we may replace ∆ by the sum
over the refined diagonals ∆θ,θ.
To get the θ-part of the decomposition for a given θ, we only take the diagonals involving
the chosen θ. Following Lemma 4.5, this requires to sum over the θ satisfying
∑ δV
δΓ
θV = δ
δΓ
θ,
and we obtain the desired identity in H•(M,Q)⊗Q[Alb(X)]. ■
In order to extract a numerical version of the refined decomposition formula which can actu-
ally be used to compute the invariant, we need deg
(
∆θ,θ,!α∩ev∗γ
)
for α∈H•
(
Mθ,Q
)
. This means
that we need expressions for the Poincaré dual classes
(
Dθ,θ
)∨
.
In the unrefined setting, an explicit expression of the classD∨ Poincaré dual to the diagonal is
provided by the Künneth decomposition, which allows us to write the classD∨ in terms of a basis
of H∗(X,Q). This provides a decomposition of ι∗D∨ but not for the
(
Dθ,θ
)∨
. Furthermore, we
do not a priori know much of the cohomology of Hθ. Therefore, this task may be especially
hard if the classes
(
Dθ,θ
)∨
are not pulled back from X.
4.4 Degeneration in the elliptic case
We first prove the following lemma.
Lemma 4.8. The diagonal components Dθ,θ indexed by θ ∈ aw/δ
(
Dθ
)
are cobordant in D (and
thus in X). In other words, ι∗
[
Dθ,θ
]
does not depend on the choice of θ ∈ aw/δ
(
Dθ
)
.
Proof. Let θ ∈ aw/δ
(
Dθ
)
and γ(t) ∈ Alb(X)|V(Γ)| be a loop based at θ. The idea is to use the
loop γ(t) to construct a cobordism between the different components Dθ,θ. We claim that there
is a unique path γ(t) ∈ Alb(X) such that γ(0) = θ and in Alb(X) we have
δ
δΓ
γ(t) =
∑ δV
δΓ
γV (t).
To construct it, let θ̃ be a lift of θ in the universal cover H0(X,ΩX)∗ of Alb(X), and consider
the loop
∑
V
δV
δΓ
γV (t) in Alb(X). We can lift the loop to a path in H0(X,ΩX)∗ starting at δ
δΓ
θ̃.
This path can be written as follows t 7→ δ
δΓ
θ̃ + ρ(t), where ρ(0) = 0 and ρ(1) ∈ H1(X,Z) ⊂
H0(X,ΩX)∗. The path ρ is determined by γ. It corresponds to a loop in Alb(X), and its class
[ρ] = ρ(1) ∈ H1(X,Z) ⊂ H0(X,ΩX)∗ satisfies [ρ] = Σ∗(γ), where Σ∗ : H1(Alb(X)|V(Γ)|,Z) →
H1(Alb(X),Z) is induced by Σ: (θ′V ) 7→
∑ δV
δΓ
θ′V . Finally, we divide by δ/δΓ: the path we care
about is the image in Alb(X) of t 7→ θ̃ + 1
δ/δΓ
ρ(t). In particular, γ(t) ∈ Alb(X) may not be
a loop.
Starting with a generic path γ(t), the path γ(t) ∈ Alb(X) is also generic. Therefore, its
preimage by aw/δ|D : D → Alb(X) provides a cobordism in D between Dθ,γ(0) and Dθ,γ(1).
26 T. Blomme and F. Carocci
To finish the proof, we need to prove that γ(1) may take any value in aw/δ
(
Dθ
)
. Equiva-
lently, we need to show that any element of Torδ/δΓ(Alb(X)) has a lift in H0(X,ΩX)∗ of the
form 1
δ/δΓ
ρ(1), with ρ as above. This amounts to the surjectivity of Σ∗. The latter is ensured
by the fact that the g.c.d. of δV /δΓ is 1, finishing the proof. ■
Remark 4.9. The key point in the proof is the surjectivity. Otherwise, the elements in the
torsor would split in different classes modulo the image of the morphism.
Remark 4.10. This lemma is easier to prove in the elliptic case since all maps are actually
group morphisms, so that the various components are in fact parallel subtori.
Now, let us work under the additional assumption that the ι∗ : H•(XnV ,Q) → H•(HθV
V ,Q
)
are surjective. This hypothesis is satisfied when X is an elliptic curve, since each H
θV
V is now
a subtorus of XnV . Using Lemma 4.8, we can now prove the following.
Lemma 4.11. Given γ ∈ H•(X,Q) and [N] a cycle class in H•
(
Mθ,Q
)
, the intersection
numbers ẽv∗[N] ∩
((
Dθ,θ
)∨ ∪ ι∗γ
)
do not depend on θ ∈ aw/δ
(
Dθ
)
:
Proof. If L is chosen generically, every choice of θ is also generic and H
θV
V is thus a submanifold
of XnV . In particular, we have Poincaré duality. Let µ be the class Poincaré dual to ẽv∗[N]
inside Hθ, so that we have
ẽv∗[N] ∩
((
Dθ,θ
)∨ ∪ ι∗γ
)
=
[
Hθ
]
∩
((
Dθ,θ
)∨ ∪ µ ∪ ι∗γ
)
=
[
Dθ,θ
]
∩ (µ ∪ ι∗γ),
since
(
Dθ,θ
)∨
is Poincaré dual to Dθ,θ inside Hθ. By surjectivity, we can write µ = ι∗µ̃.
Moreover, as we compute an intersection number, we may as well compute its push-forward
inside H0(X,Q) and use push-pull formula, so that the number we care about is
ι∗
([
Dθ,θ
]
∩ ι∗(µ̃ ∪ γ)
)
= ι∗
[
Dθ,θ
]
∩ (µ̃ ∪ γ).
As by Lemma 4.8 ι∗
[
Dθ,θ
]
does not depend on θ, we get that the intersection numbers do not
depend on the particular choice of θ ∈ aw/δ
(
Dθ
)
. ■
We may now state the degeneration formula under the surjectivity assumption.
Theorem 4.12. Consider a one parameter family degeneration Yt of P1-bundles with common
base X whose central fiber is a union of P1-bundle glued over their boundary divisors. Under
the surjectivity assumption, we have the following decomposition of the virtual classes:
[[M0]]
δ ≡
∑
Γ
∑
θ
∏
we
|Aut(Γ)|
(
∆!
∏
V
[
M
θV
V
]vir)
d
[
1
δ/δΓ
](∑ δV
δΓ
θV
)
,
where ∆: D → X is the usual diagonal inclusion.
The “≡” means that the equality is true when we compute the intersection with the pullback
of a cohomology class by the evaluation map.
Proof. We just rebrand Theorem 4.7 using Lemma 4.11 to show that both classes have the
same intersection numbers with classes of the form ι∗γ. Notice that since this is a statement
about intersection numbers, as in equation (4.2), we replace the sum over labeled graphs Γ̃ by
a sum over degeneration graphs Γ, each one having precisely |Eb(Γ)|!
|Aut(Γ) labellings. Taking into
account the degree of ϕΓ and the virtual degree of F gives the coefficient for each Γ, which is
independent of the refinement.
Correlated Gromov–Witten Invariants 27
Let
[
Mθ
]vir
=
∏[
M
θV
V
]vir
and r be the dimension of Alb(X), so that the cardinality of δ/δΓ-
torsion elements is (δ/δΓ)
2r. Intersection numbers in the (co)homology of Hθ between a class
in H•
(
Hθ,Q
)
, some
(
Dθ,θ
)∨
and some ι∗γ can be computed in the (co)homology of X by push-
pull-formula. By Lemma 4.11, the results do not depend on the specific choice of θ. Therefore,
they are equal to their average
ι∗
(
ẽv∗
[
Mθ
]vir ∩ ((Dθ,θ
)∨ ∪ ι∗γ
))
= ι∗
(
ẽv∗
[
Mθ
]vir ∩( 1
(δ/δΓ)2r
∑
θ
(
Dθ,θ
)∨ ∪ ι∗γ
))
=
1
(δ/δΓ)2r
ι∗
(
ẽv∗
[
Mθ
]vir ∩ ι∗D∨ ∪ ι∗γ
)
=
1
(δ/δΓ)2r
ev∗
[
Mθ
]vir ∩ (D∨ ∪ γ
)
.
As the 1
(δ/δΓ)2r
are precisely the coefficients appearing through the use of d
[
1
δ/δΓ
]
and its support
the θ we care about, we get the result. ■
Corollary 4.13. Under the surjectivity assumption, the full correlated virtual classes satisfy the
following decomposition formula:
[[M0]]
δ ≡
∑
Γ
∏
we
|Aut(Γ)|
d
[
1
δ/δΓ
]
∆!
∏
V
[[MV ]]
δΓ .
Proof. We rewrite Theorem 4.12 to make appear the full refined classes at the vertex level∏
V
[[MV ]]
δΓ =
∏
V
m
[
δV
δΓ
](∑
θV
[
M
θV
V
]vir · (θV )) =
∏
V
∑
θV
[
M
θV
V
]vir · (δV
δΓ
θV
)
=
∑
θ
∏
V
[
M
θV
V
]vir · (∑ δV
δΓ
θV
)
,
yielding the result. ■
Remark 4.14. The formula tells us that even if each vertex is entitled to a refinement at the
level δV , to recover the refinement for the global degeneration graph at the level δ, we only need
to know the refinements at the level δΓ. Moreover, due to the presence of d
[
1
δ/δΓ
]
, the class
associated to the graph Γ is invariant under Torδ/δΓ(Alb(X)). In particular, if δΓ = 1, all the
correlated classes coming from Γ yield the same invariants.
5 Computation of local invariants
5.1 General considerations and statement
Our goal in this section is to compute the correlated GW invariants in the case where X = E
is an elliptic curve, the genus of the source curve is g = 1, and with a unique interior point
constraint. These local invariants will be used in combination with Theorem 4.12 to obtain an
explicit computation algorithm and derive regularity results for general correlated invariants in
Section 6.
We consider Y = P(O⊕ L) for L ∈ Pic0(E); fix an homology class β = a[E] ∈ H2(E,Z) and
a vector w = (w1, . . . , wn) of tangency orders. The moduli space
M(a,w) = M1,1
(
Y |D±, a[E],w
)
28 T. Blomme and F. Carocci
is the moduli space of log stable maps as in the previous section. For δ|gcd(wi), we saw that
it decomposes into components Mθ(a,w). The correlators θ satisfy δθ ≡ φa[E](L) = aλ,
where λ = φ[E](L) is the image of the line bundle L ∈ Pic0(E) ≃ E through the isomor-
phism φ[E]. We denote the torsor by T δ(L, a).
5.1.1 Uncorrelated case
We recall the computation of the non-refined invariant
⟨pt0, 1w1 ,ptw2
, . . . ,ptwn
⟩1,a[E],w =
∫[
M(a,w)
]vir ev∗0(pt) n∏
2
ev∗i (pt).
Lemma 5.1. The uncorrelated relative GW invariant has the following value:
⟨pt0, 1w1 ,ptw2
, . . . ,ptwn
⟩1,a[E],w = an−1σ(a) · w2
1,
where σ(a) =
∑
d|a d is the sum of divisors function.
Proof. This elementary formula is computed, for instance, in [7]. Briefly, it may be obtained
as follows. Genus 1 parametrized curves in Y realizing the class a[E] + b(w)
[
P1
]
come from
degree a covering maps f : C → E. Up to translation, these are group homomorphisms. We may
fix the translation parameter using the marked point p0, mapped to a fixed point in E. Then
we see that these are in bijection with the index a sublattices of π1(E) ≃ Z2, of which there
are σ(a).
Given one of the σ(a) covers f : C → E, enhancing to a map to Y = P(O ⊕ L) amounts to
find a section of f∗L, with poles and zeros of prescribed order wi and fixed image in E.
We denote by yi the marked points in C (with not trivial contact order) and let xi = f(yi)
their image in E. For 2 ⩽ i ⩽ n, there are a possible choices for each yi given that xi is fixed.
The position of y1 is determined by the relation
∑
wiyi ≡ f∗(λ) in C, i.e., y1 is a w1-root
of f∗(λ)−
∑n
i=2wiyi. There are w2
1 possible choices for y1, yielding the result. ■
5.1.2 Statement
We now consider the correlated invariants
⟨⟨pt0, 1w1 ,ptw2
, . . . ,ptwn
⟩⟩δ1,β,w =
∫
[[M(a,w)]]δ
ev∗0(pt)
n∏
2
ev∗i (pt),
which is an element of the group algebra Q[E] with support on T δ(L, a).
Before giving a closed formula for the full local correlated invariant, we need to introduce
certain functions with values in Q[E].
First, we consider the average of torsion elements
ϑd :=
1
d2
∑
dθ≡0
(θ) ∈ Q[E].
They satisfy ϑd1ϑd2 = ϑlcm(d1,d2). We then define the following functions: for d|δ,
ϑδ(d) =
∏
p
(ϑpνp(d) − 1νp(d)<νp(δ)ϑpνp(d)+1),
where the products is over primes and νp is the p-adic valuation. The 1 indicates that the second
term for each factor of the product only appears if νp(d) < νp(δ).
Correlated Gromov–Witten Invariants 29
Example 5.2. If d = δ, so that ϑδ(δ) = ϑδ. If δ = pv is the power of a prime number p, the
values of ϑpv for its divisors 1, p, . . . , pv−1, pv are ϑ1 − ϑp, ϑp − ϑp2 , . . . , ϑpv−1 − ϑpv and ϑpv .
Next, we consider the variants of the arithmetic function σ(a) =
∑
d|a d defined as follows:
σd(a) = σ(a/d) if d|a and 0 else. Combining both, we define the following function, with values
in Q[Torδ(E)]
σδ(a) =
∑
d|δ
σδ/d(a)ϑδ(d).
For concrete computations, it may be convenient to express σδ(a) as a linear combination of
the ϑd for d|δ but with different coefficients. To do so, consider the following function on N:
Υδ
d(a) =
∏
p
(
σp
νp(d) − 1νp(d)<νp(δ)σ
pνp(d)+1)(
pνp(a)
)
.
In the above product of function, the argument a is factorized over prime numbers, so that
for a =
∏
pνp(a) and d =
∏
pνp(d),
σd(a) =
∏
σp
νp(d)(
pνp(a)
)
.
The following identity is proven in the proof of Theorem 5.3. Thanks to multiplicativity, it
suffices to check it for powers of primes, where it stems from a summation by part
σδ(a) =
∑
d|δ
Υδ
d(a)ϑδ/d.
Doing a summation by parts for each prime numbers is called a multiplicative summation by
parts.
The torsor consists of δ-roots of aλ. However, aλ has no canonical δ-root. The furthest we
can naturally define is the gcd(a, δ)-root aλ
gcd(a,δ) . Then, pick θ0 to be any choice of root such
that δ
gcd(a,δ)θ0 =
a
gcd(a,δ)λ. A correlator θ0 as before is called a special correlator. Theorem 3.11
ensures that the result does not depend on the choice of the latter.
One way to construct special correlators is as follows: if λ0 satisfies δλ0 = λ, one may
take θ0 = aλ0. Indeed, one has
δ
gcd(a, δ)
(aλ0) =
a
gcd(a, δ)
δλ0 =
a
gcd(a, δ)
λ.
Theorem 5.3. The full local correlated invariant has the following expression:
⟨⟨pt0, 1w1 , ptw2
, . . . ,ptwn
⟩⟩δ1,a[E],w = an−1w2
1σδ(a) · (θ0), (5.1)
where θ0 is any special correlator, i.e., satisfies δ
gcd(a,δ)θ0 =
a
gcd(a,δ)λ.
Without assuming Theorem 5.3, it can be checked from the definition of σ that
σδ(a) = σδ(a)ϑδ/ gcd(a,δ), (5.2)
so that the right-hand side of (5.1) does not depend on which θ0 we choose. It may thus be
replaced by ϑδ/ gcd(a,δ) · (θ0) = d
[
1
δ/ gcd(a,δ)
](
a
gcd(a,δ)λ
)
.
Assuming instead Theorem 5.3, equation (5.2) is in fact a consequence from Theorem 3.11,
which tells us that the correlated invariant is invariant when multiplying by an element of
φa[E](Torδ(E)) = a · Torδ(E) = Tora/ gcd(a,δ)(E).
See the second part of the proof of Lemma 5.7 for a proof of the second equality. In particular,
it is also invariant by multiplication by ϑδ/ gcd(a,δ).
With the above formulation, the correlated invariant appears as a refined version of the
uncorrelated invariant, and the refinement takes the form of σδ replacing σ.
30 T. Blomme and F. Carocci
5.1.3 Applications
Before going to the proof, we present some applications.
Example 5.4. If a is coprime with δ, every σδ/d(a) vanishes except for d = δ, and we thus recover
that σδ(a) = σ(a)ϑδ. In other words, the curves are equally spread among the correlators.
Example 5.5. Let J2 be the second Jordan function, defined by J2(p
α) = p2α−2
(
p2 − 1
)
. It
counts the number of elements of order n in (Z/nZ)2.
The (0)-coefficient of ϑδ(d) is equal to the product of (0)-coefficients for each prime p. The lat-
ter is equal to 1
p2νp(d)
− 1
p2νp(d)+2 if νp(d)<νp(δ) and
1
p2νp(δ)
else. This matches the values of the func-
tion J2(δ/d)
δ2
, which is also multiplicative. Therefore, we get that
⟨pt0, 1w1 ,ptw2
, . . . ,ptwn
⟩θ01,a[E],w = an−1
(w1
δ
)2∑
d|δ
J2(d)σ
d(a).
However, this application is partially a lie since this computation is actually the first step toward
proving Theorem 5.3.
From m [δ/δ′]
(
[[M(a,w)]]δ
)
= [[M(a,w)]]δ
′
, we deduce that the functions σδ satisfy
m
[
δ/δ′
]
(σδ(a)) = σδ′(a),
which may also be checked directly from the definition of σδ.
Using Theorem 5.3, we immediately get the quasi-modularity result for the generating series
of correlated invariants. Here, we extend the notion of quasi-modularity for functions with value
in a vector space, here chosen to be C[E]. In the finite-dimensional case, it just means that all
coordinate functions are quasi-modular forms. See Section 6.3 for more details.
Corollary 5.6. The following generating series is quasi-modular for Γ0(δ)∑
a
⟨⟨pt0, 1w1 ,ptw2
, . . . ,ptwn
⟩⟩δ1,a[E],wq
a.
Proof. The generating series of σ is the first Eisenstein series E2(q), known to be a quasi-
modular form. The result thus follows from the quasi-modularity of the generating series∑
a σ
d(a)qa = E2(q
d) for the congruence subgroup Γ0(d). ■
The rest of the section is dedicated to the proof of Theorem 5.3, by refining the proof of
Lemma 5.1. We proceed in several steps. The first is to study the curves coming from a common
covering map f : C → E. Summing over covering maps in Section 5.3, we are then able to find
a closed expression for the (θ0)-coefficient. Multiplicativity properties and an induction relation
are thus sufficient to prove the formula from Theorem 5.3.
5.2 Contribution of a fixed cover
Let us consider a fixed cover f : C → E, which is a group homomorphism choosing the marked
point p0 and its image as neutral element. Its kernel ker f has cardinality a, the degree of the
covering. Let f∗ : E → C be the dual map between the curves seen as their Picard groups. The
lifting condition writes itself
∑n
1 wjyj = f∗(λ), where λ ∈ E corresponds to the line bundle L.
We have the following group morphism:
Cn −→ En−1 × C, (kj) 7−→
(
f(k2), . . . , f(kn),
n∑
1
wjkj
)
Correlated Gromov–Witten Invariants 31
with its kernel K(f) =
{
(kj) ∈ C × (ker f)n−1 s.t.
∑n
1 wjkj = 0
}
. In particular, it sits in the
following exact sequence, from which we see it has cardinality w2
1a
n−1
0 → Torw1(C) → K(f) → (ker f)n−1 → 0.
For the cover f , the set of curves matching the constraints (xj) is in bijection with the
following K(f)-torsor
Sλ,x2,...,xn =
{
(yj) ∈ Cn s.t. ∀ 2 ⩽ j ⩽ n f(yj) = xj and
n∑
1
wjyj = f∗(λ) ∈ C
}
,
also having cardinality w2
1a
n−1. We now wish to refine the above description. To do so, we use
the correlator function
κδ : Cn −→ E, (yj) 7−→
n∑
1
wj
δ
f(yj).
The correlators are the elements θ ∈ E satisfying δθ = f∗f
∗(λ) = aλ. Among them, recall
we have the special correlators satisfying δ
gcd(a,δ)θ0 = a
gcd(a,δ)λ and that aλ0, where δλ0 = λ is
a special correlator.
Lemma 5.7. The image of K(f) via the correlator function is κδ(K(f)) = f(Torδ(C)). Fur-
thermore, it contains Torδ/ gcd(a,δ)(E).
Proof. The equality follows from the definitions. Assume that (kj) ∈ K(f). Then we have that∑n
1 wjkj = 0 and it follows that
∑n
1
wj
δ kj ∈ Torδ(C) and consequently κδ((kj)) = f
(∑n
1
wj
δ kj
)
∈
f(Torδ(C)). Conversely, let f(s) ∈ f(Torδ(C)) with s ∈ Torδ(C). Let k1 be such that s = w1
δ k1,
which exists because C is divisible, and kj = 0 ∈ ker f if j ⩾ 2. We have
f(s) = f
(w1
δ
k1
)
= κδ(k1, 0, . . . , 0) and
n∑
1
wjkj = δs = 0.
Thus, we have the reverse inclusion f(Torδ(C) ⊂ κδ(K(f)).
We now prove that Torδ/ gcd(a,δ)(E) ⊂ f(Torδ(C)). Using the dual morphism f∗, we start
from f∗(Torδ(E)) ⊂ Torδ(C). We now apply the morphism f and use that the composition
f ◦ f∗ : E → E is the multiplication by a, so that a · Torδ(E) ⊂ f(Torδ(C)). We now claim
that a · Torδ(E) = Torδ/ gcd(a,δ)(E), which concludes the proof:
� If x is δ-torsion, we have δ
gcd(a,δ)ax = a
gcd(a,δ)δx = 0, so that we have the inclusion
a · Torδ(E) ⊂ Torδ/ gcd(a,δ)(E).
� As |Toru(E)| = u2, the short exact sequence
0 → Tora(E) ∩ Torδ(E) = Torgcd(a,δ)(E) → Torδ(E)
a·−→ a · Torδ(E) → 0,
ensures that they have the same cardinality
(
δ
gcd(a,δ)
)2
. ■
The following proposition gives a description of the correlators in the image of the tor-
sor Sλ,x2,...,xn .
Proposition 5.8. The correlators achieved by the solutions, i.e., the set κδ(Sλ,x2,...,xn), form
a f(Torδ(C))-torsor which contains the special correlators θ0. Solutions are uniformely spread
among the correlators.
32 T. Blomme and F. Carocci
Proof. The solutions form the K(f)-torsor Sλ,x2,...,xn . Applying the correlator function, we
immediately get a torsor under κδ(K(f)), equal to f(Torδ(C)) by Lemma 5.7. We also get that
the solutions split evenly among the elements.
We now need to prove that it contains the special correlators. Special correlators form
a torsor under Torδ/ gcd(a,δ)(E), which lies in f(Torδ(C)) by Lemma 5.7. Thus, it suffices
to show it contains one of them. Pick y2, . . . , yn such that f(yj) = xj and choose y1 such
that
∑n
1
wj
δ yj = f∗(λ0), with δλ0 = λ; notice that this always exists as C as well is a divisible
group. In particular, as δλ0 = λ, we have that
∑n
1 wjyj = f∗(λ) and (yj) is indeed an element
of Sλ,x2,...,xn . Moreover, we have that κδ((yj)) = f ◦ f∗(λ0) = aλ0, which is one of the special
correlators. Therefore, all special correlators belong to the image torsor. ■
To finish this section, we provide a first expression of the local correlated invariant as a sum
over the covers f : C → E. The cover f corresponds to a sublattice Λ ⊂ π1(E) ≃ Z2. We can
find a basis (e1, e2) such that Λ = ⟨ke1, (a/k)e2⟩ for some unique k such that k2|a. We say
that Λ, and by extension the associated cover, is of type (k, a/k).
Let ϑ(f) ∈ Z[E] be the element with coefficient 1 for every element in f(Torδ(C)).
Proposition 5.9. The correlated invariant admits the following expression:
⟨⟨pt0, 1w1 , ptw2
, . . . ,ptwn
⟩⟩δ1,a[E],w = an−1
(w1
δ
)2 ∑
f : C→E
gcd(k, δ) gcd(a/k, δ)ϑ(f) · (θ0).
Proof. We know by Proposition 5.8 that the w2
1a
n−1 solutions for a fixed cover f : C → E
uniformely spread among the possible correlators, so that we get some integer multiple of the
torsor ϑ(f) · (θ0) indexing the possible correlators. To conclude, we merely need to compute the
cardinality of f(Torδ(C)).
The basis (e1, e2) diagonalizing the lattice inclusion provides real coordinates on E and C
such that f has the following expression:
f : C ≃ (R/Z)2 −→ E ≃ (R/Z)2, (u, v) 7−→ (ku, (a/k)v)
from which we see that |f(Torδ(C))| = δ2
gcd(k,δ) gcd(a/k,δ) . Therefore, each correlator carries
an−1
(w1
δ
)2
gcd(k, δ) gcd(a/k, δ)
solutions. Summing over covers yields the result. ■
Our next goal is to find an expression avoiding the summation over the morphisms f : C → E,
i.e., the index a sublattices of π1(E) ≃ Z2.
5.3 Computation of the (θ0)-coefficient
The next step toward the proof of Theorem 5.3 is to use Proposition 5.9 to compute the (θ0)-
coefficient, where θ0 is a special correlator, belonging to the support of all terms in the sum.
To do so, we need to know the number of sublattices of π1(E) ≃ Z2 of a given type, which is
precisely the definition of the Dedekind ψ-function: it the unique multiplicative function such
that for any prime number p, ψ(pα) = pα−1(p + 1). There are ψ(n) index n sublattices of Z2
having type (1, n), which are also known as primitive sublattices. Merely dividing by k, we get
that there are ψ
(
a
k2
)
sublattices of type (k, a/k).
Example 5.10. We have ψ(2) = 3 as there are 3 sublattices of Z2 of index 2. If (e1, e2) is
a basis of Z2, these lattices are ⟨2e1, e2⟩, ⟨e1, 2e2⟩ and ⟨e1 + e2, e1 − e2⟩.
Correlated Gromov–Witten Invariants 33
Remark 5.11. In particular, as there are σ(a) index a sublattices, we get that∑
k2|a
ψ
( a
k2
)
= σ(a).
This may also be checked using the multiplicativity of ψ and σ and the fact that the identity is
true on powers of prime numbers.
Using Proposition 5.9, we get the following expression.
Lemma 5.12. Let sδ(a) =
∑
k2|a gcd(k, δ) gcd(a/k, δ)ψ
(
a
k2
)
. For the level δ refinement, the (θ0)-
coefficient is equal to an−1
(
w
δ
)2
sδ(a).
Proof. We need to compute the (0)-coefficient of
∑
f : C→E gcd(k, δ) gcd(a/k, δ)ϑ(f). As each ϑ
has by definition coefficient 1 at (0), and as there are ψ
(
a
k2
)
lattices of type (k, a/k), we get the
result. ■
Before getting to the computation of the full local correlated invariant, we transform the
above expression to better suit for our purposes, transforming the sum over k2|a into a sum
over d|δ using arithmetic functions more convenient than ψ.
� We already introduced the sum of divisors function σ(a) =
∑
d|a d, whose Dirichlet gener-
ating series is ζ(s)ζ(s− 1).
� Given d ∈ N, the Dirichlet generating series of σd(a) = 1d|aσ(a/d) is
∞∑
a=1
σd(a)
as
=
∞∑
a′=1
σ(a′)
(da′)s
=
1
ds
ζ(s)ζ(s− 1).
� The Dirichlet generating series of the second Jordan function J2 is
∞∑
d=1
J2(d)
ds
=
ζ(s− 2)
ζ(s)
.
Lemma 5.13. The function sδ(a) is fully multiplicative in the following sense:
sδ(a) =
∏
spνp(δ)
(
pνp(a)
)
,
where products are over the set of prime numbers. Furthermore, we have the following expression:
sδ(a) =
∑
d|δ
J2(d)σ
d(a).
Proof. We first prove the multiplicativity: assume that we have the decomposition in products
of primes a =
∏
pαp and δ =
∏
pδp . The sum over k2|a writes itself as sum over the families (ip)
with 2ip ⩽ αp indexed by the set of prime numbers P∑
k2|a
gcd(k, δ) gcd(a/k, δ)ψ
(
a/k2
)
=
∑
2ip⩽αp
∏
ψ
(
pαp−2ip
)
pmin(ip,δp)+min(αp−ip,δp),
using the multiplicativity of the Dedekind function ψ. We can factor the sum as a product and
get the sought multiplicativity.
34 T. Blomme and F. Carocci
To prove the desired identity, we compute both Dirichlet generating series. The multiplica-
tivity allows to factor the generating series as a product over prime numbers so that
∞∑
a,δ=1
sδ(a)
δsat
=
∏
p∈P
( ∑
d,2i⩽α
ψ
(
pα−2i
)
pmin(i,d)+min(α−i,d)p−sd−tα
)
.
To finish the computation of the Dirichlet series for sδ(a), the inner sum may be rewritten∑
i,j,k
ψ
(
pi
)
pmin(j,k)+min(i+j,k)p−ks−2jt−it,
where p is a prime number and i, j, k ⩾ 0. Setting aside i = 0, ψ
(
pi
)
= pi−1(p + 1) and we
can split according to the value of k and compute the geometric sums, yielding some rational
function in p−t and p−s.
For the second expression, the Dirichlet series is
∑
δ,a
∑
d|δ J2(d)σ
d(a)
δsat
=
∑
d,k,a
J2(d)σ
d(a)
dsksat
(where δ = dk)
=
∑
d,k
J2(d)
ds
1
ks
1
dt
ζ(t)ζ(t− 1) =
ζ(s+ t− 2)
ζ(s+ t)
ζ(s)ζ(t)ζ(t− 1).
Using that ζ(s) =
∏ 1
1−p−s , it can also be expressed as the product of values of a rational
function at prime numbers. By a technical but elementary computation, we check that both
rational functions are actually the same, so that generating functions have equal coefficients,
finishing the proof of the identity. ■
Notice that if gcd(a, δ) = 1, we have σd(a) = 0 for every divisor d of δ. Then, we have
sδ(a) = σ(a). This way, the remaining terms may appear as a correction term when gcd(a, δ) ̸= 1.
Example 5.14. If p, q are distinct prime numbers, we have the following values:
sp = σ +
(
p2 − 1
)
σp, sp2 = σ +
(
p2 − 1
)
σp +
(
p4 − p2
)
σp
2
,
spq = σ +
(
p2 − 1
)
σp +
(
q2 − 1
)
σq +
(
p2 − 1
)(
q2 − 1
)
σpq.
5.4 Case of other correlators
To finish the proof of Theorem 5.3, we now study the case where θ ̸= θ0 is another correlator.
We proceed in several steps:
� First prove that the coefficients do not depend on the precise correlator θ but only on the
order of θ − θ0 inside Torδ(E). To do so, we use the deformation invariance for a suitable
family where we deform the base curve E.
� Then, through Lemma 3.12, we prove an induction relation that enables a formal compu-
tation of the correlated invariant for θ ̸= θ0.
� Using the induction relation, we show the multiplicativity of the coefficient functions, so
that we may restrict to the case of powers of primes, which we compute also using the
induction. Combination of both leads to Theorem 5.3.
Correlated Gromov–Witten Invariants 35
5.4.1 Dependence on the order
Assumed that L = O and choose θ0 = 0, so that the torsor of correlators is actually Torδ(E).
For θ ∈ Torδ(E), its order ω(θ) is the smallest positive integer n such that nθ ≡ 0.
Lemma 5.15. Assuming L = O, the correlated invariant ⟨pt0, 1w1 , ptw2
, . . . ,ptwn
⟩θ1,a[E],w only
depends on the order ω(θ).
Proof. Write E as some Eτ = C/⟨1; τ⟩ for some complex number τ in the Poincaré half-plane.
For any γ =
(
a b
c d
)
∈ SL2(Z), we have an isomorphism between Eτ and Eγ·τ where γ · τ = aτ+b
cτ+d .
The isomorphism is actually given by
φ : (zmod1, τ) 7−→ z
cτ + d
mod1, γ · τ.
Choose a path τ(t) in the Poincaré half-plane going from τ(0) = τ to τ(1) = γ · τ . The δ-torsion
elements in Eτ(t) are deformed continuously along with τ(t) as they can be written as uτ(t)+ v,
where u, v ∈ Torδ(R/Z). The identification φ : Eτ → Eγ·τ then induces some monodromy among
torsion elements. More precisely, if u, v ∈ Torδ(R/Z), we have
φ−1
(
u
aτ + b
cτ + d
+ v
)
= u(aτ + b) + v(cτ + d) = (au+ cv)τ + (bu+ dv).
Therefore, we deduce that the values of the correlated invariants for the torsion elements θ =
uτ + v and γ · θ = (au + cv)τ + (bu + dv) are the same. As the action of SL2(Z) on the set of
torsion elements of the same order in Torδ
(
(R/Z)2
)
is transitive, we conclude. ■
5.4.2 Coefficients, induction relation and multiplicativity
For r|δ, let sδ[r](a) be such that for any θ ∈ Torδ(E),
⟨pt0, 1w, ptw2
, . . . ,ptwn
⟩θ1,a[E],w = an−1
(w
δ
)2
sδ[ω(θ)](a).
In particular, we have sδ[1] = sδ, corresponding to θ = 0. We also set the global function with
values in the group algebra Sδ(a) =
∑
δθ≡0 sδ[ω(θ)](a) · (θ), so that
⟨⟨pt0, 1w,ptw2
, . . . ,ptwn
⟩⟩δ1,a[E],w = an−1
(w
δ
)2
Sδ(a).
The definition of Sδ directly comes from the invariants. Theorem 5.3 consists in proving that
Sδ
δ2
= σδ, as defined before Theorem 5.3.
Lemma 5.16. For every pair δ′|δ, we have the following relation:∑
r|δ′
J2(r)sδ[r](a) = (δ′)2sδ/δ′(a).
Proof. We already have m [δ′]
(
[[M(a,w)]]δ
)
= [[M(a,w)]]δ/δ
′
. Integrating the point constraints
and looking at the (0)-coefficient yields
an−1
(
w
δ/δ′
)2
sδ/δ′(a) =
∑
δ′θ≡0
an−1
(w
δ
)2
sδ[ω(θ)](a).
The sum is over θ such that δ′θ ≡ 0. For any r|δ′, there are J2(r) elements of order exactly r,
yielding the desired relation. ■
36 T. Blomme and F. Carocci
These relations form a system which is triangular for the order given by the divisibility. They
are thus sufficient to compute all the functions sδ[r]. We carry out some examples below.
Example 5.17. If δ = p is a prime number, we have sp[1] +
(
p2 − 1
)
sp[p] = p2σ. As we already
know that sp[1] = sp = σ +
(
p2 − 1
)
σp, we deduce that(
p2 − 1
)
σp +
(
p2 − 1
)
sp[p] =
(
p2 − 1
)
σ,
and thus sp[p] = σ − σp.
Example 5.18. For δ = p2, we have the following equations:
sp2 +
(
p2 − 1
)
sp2 [p] +
(
p4 − p2
)
sp2
[
p2
]
= p4s1 = p4σ,
sp2 +
(
p2 − 1
)
sp2 [p] = p2sp = p2σ +
(
p4 − p2
)
σp,
sp2 = sp2 = σ +
(
p2 − 1
)
σp +
(
p4 − p2
)
σp
2
,
which solves for
sp2 = σ +
(
p2 − 1
)
σp +
(
p4 − p2
)
σp
2
, sp2 [p] = σ +
(
p2 − 1
)
σp − p2σp
2
,
sp2 [p
2] = σ − σp.
To reduce the computation down to the powers of primes, we use the induction relation to
prove the multiplicativity of the functions sδ[r](a) and Sδ(a).
Proposition 5.19. The function sδ[r](a) and Sδ(a) are fully multiplicative: for pairs of coprime
elements (r1, r2), (a1, a2) and (δ1, δ2), we have
sδ1δ2 [r1r2](a1a2) = sδ1 [r1](a1)sδ2 [r2](a2) and Sδ1δ2(a1a2) = Sδ1(a1)Sδ2(a2).
Proof. We show multiplicativity by induction. It is true for r = 1 by multiplicativity of sδ.
For the induction step, we use the identity from Lemma 5.16 and the multiplicativity of sδ,
(δ′1)
2(δ′2)
2sδ1/δ′1(a1)sδ2/δ′2(a2) = (δ′1δ
′
2)
2sδ1δ2/δ′1δ′2(a1a2),∑
r1|δ′1
r2|δ′2
J2(r1)J2(r2)sδ1 [r1](a1)sδ2 [r2](a2) =
∑
r|δ′1δ′2
J2(r)sδ1δ2 [r](a1a2)
=
∑
r1|δ′1
r2|δ′2
J2(r1)J2(r2)sδ1δ2 [r1r2](a1a2).
To get the last sum, we use that each divisor of δ′1δ
′
2 can uniquely been written as a product of
a divisor of δ′1 and a divisor of δ′2. If we assume multiplicativity for r a strict divisor of δ′1δ
′
2,
all the terms in the sum except the last one are equal, and we are left with the multiplicativity
for r = δ′1δ
′
2, finishing the induction. The multiplicativity of Sδ follows from the multiplicativity
of sδ[r],( ∑
δ1θ1≡0
sδ1 [ω(θ1)](a1) · (θ1)
)( ∑
δ2θ2≡0
sδ2 [ω(θ2)](a2) · (θ2)
)
=
∑
δ1θ1≡δ2θ2≡0
sδ1 [ω(θ1)](a1)sδ2 [ω(θ2)](a2) · (θ1 + θ2).
As each δ1δ2 torsion element can be written uniquely as the sum of a δ1-torsion and a δ2-torsion
element, we conclude. ■
Correlated Gromov–Witten Invariants 37
5.4.3 Computation for powers of primes
Because of multiplicativity, we only need to compute the function when a, r, δ are the power of
a common prime p. Recall that we defined the elements of Q[E]: ϑd = 1
d2
∑
dθ≡0(θ). We also
momentarily set ϑprimd =
∑
ω(θ)=d(θ).
Proposition 5.20. The function (δ, r, a) 7→ sδ[r](a) has the following values over powers of
primes, for r > 0
spd [p
r] = spd−r − p2d−2rσp
d−r+1
=
d−r∑
0
J2
(
pj
)
σp
j − p2d−2rσp
d−r+1
.
Consequently, we find the following expressions for Spd:
Spd
p2d
=
d−1∑
r=0
(
σp
r − σp
r+1)
ϑpd−r + σp
d
ϑ1 = σϑpd +
d∑
1
σp
r
(ϑpd−r − ϑpd−r+1).
Proof. By Lemma 5.16, we have the following relations:
p2rspd−r =
r∑
j=0
J2
(
pj
)
spd
[
pj
]
.
For r > 0, making the difference between relations for r and r − 1, we get
p2rspd−r − p2r−2spd−r+1 = J2(p
r)spd [p
r] = p2r−2
(
p2 − 1
)
spd [p
r].
Therefore, after dividing by p2r−2, using that spd−r+1 = spd−r + J2
(
pd−r+1
)
σp
d−r+1
, we finally
get (
p2 − 1
)
spd−r − J2
(
pd−r+1
)
σp
d−r+1
=
(
p2 − 1
)
spd [p
r].
Dividing by p2 − 1 yields the desired expression. Then, we deduce that Spd has the following
expression:
Spd = spdϑ1 +
d∑
r=1
(
spd−r − p2(d−r)σp
d−r+1) · ϑprimpr
= spdϑ1 +
d∑
r=1
(
spd−r − p2(d−r)σp
d−r+1) · (p2rϑpr − p2r−2ϑpr−1
)
.
We can now split the sum performing a summation by part to get
Spd =
(
spd − spd−1 + p2(d−1)σp
d)
ϑ1 +
(
σ − σp
)
p2dϑpd
+
d−1∑
1
p2rϑpr
(
spd−r − p2(d−r)σp
d−r+1 − spd−r−1 + p2(d−r−1)σp
d−r)
= p2dσp
d
ϑ1 + p2d
d∑
1
(
σp
d−r − σp
d−r+1)
ϑpr .
This yields the first expression. Performing a second summation by part yields the second
expression. ■
38 T. Blomme and F. Carocci
We have two expressions of Spd , depending on which feature we wish to emphasize: the ϑd
or the σd. The first ones are useful to compute in the group algebra since they are projectors,
while the second are multiplicative functions that provide quasi-modularity properties.
Proof of Theorem 5.3. Combining the multiplicativity from Proposition 5.19 and the ex-
pression for powers of primes from Proposition 5.20 yields and explicit expression for Sδ and we
finally get that Sδ
δ2
= σδ. ■
6 Floor diagrams and regularity
Our goal in this section is to obtain regularity results in the flavor of [7] in the refined setting
provided by the correlation. To do so, we use the correlated degeneration formula from The-
orem 4.12 along with the local computation from Theorem 5.3. The use of the decomposition
formula for a family of E × P1 with central fiber a chain of E × P1 glued along their boundary
divisors leads to combinatorial objects called floor diagrams. The latter appear in [7] although
they are obtained through tropical techniques. Using these floor diagrams, we are able to prove
the quasi-modularity in the base direction, and the piecewise polynomiality in the tangency
orders. We explain how to adapt the floor diagram to the correlated setting, using coefficients
in the group algebra Q[E].
6.1 Floor diagrams and their multiplicities
We first recall the notion of floor diagram in the setting of curves in E × P1, already developed
in [7].
Definition 6.1. A floor diagram D is the data of a weighted oriented graph with the following
properties:
(1) The graph has three kind of vertices:
� sinks and sources which are univalent vertices, referred as infinite vertices,
� flat vertices which are bivalent with one ingoing and one outgoing edge,
� floors which carry a label aV ∈ N.
(2) The set of flat vertices and floors carry a total order compatible with the orientation.
(3) The edges have a positive weight we, such that the weighting makes floors and flat vertices
balanced.
(4) The complement of all flat vertices is without cycle and each connected component of the
complement contains a unique sink or source.
For floor diagram, as for curves, we have a notion of genus and class, recalled in the next
definition.
Definition 6.2. Let D be a floor diagram.
� The genus g(D) is its genus as a graph where floors are considered to have genus 1:
g(D) = b1(D) + |V (D)|, where V (D) is the set of floors.
� The class of D is a[E] + b
[
P1
]
where a =
∑
V aV is the sum of floors weights, and b is the
sum of flows at the sources (or sinks, since the flow through the diagram is preserved by
the balancing condition).
� The tangency profile w is the collection of ±we for edges adjacent to infinite vertices (with
a + for sinks and − for sources).
� The δ-gcd of a diagram is the gcd of its edge weights and δ.
Correlated Gromov–Witten Invariants 39
CORRELATED GROMOV-WITTEN INVARIANTS 41
•
•
•
•2 2
4
•2
2
•
2 •
•
•
•
3 •3
6
•6
4•8
•6
•3 3
•
6
D1 D2
Figure 2. Floor diagrams.
Theorem 6.5, the latter are exactly the graphs appearing in the decomposition formula, up
to a redecoration.
▷ Sources and sinks correspond to marked points mapped to the boundary divisors of
the expanded degeneration.
▷ Flat vertices correspond to genus 0 components with an internal marked point mapped
to a fiber of E × P1.
▷ Floors correspond to genus 1 components in the class aV [E] + bV [P1] where bV is the
flow through V . Its intersection profile with the boundary divisor is prescribed by the
adjacent edges.
▷ Edges correspond to nodes in the domain mapped to the singular locus of the expanded
degeneration.
▷ Edges relating vertices which are not consecutive in the order on floors and flat vertices
should be subdivided. The new bivalent vertices would correspond to components
mapped to a fiber but without marked point.
Let w = (w1, . . . , wn) be a tangency profile and let δ be a common divisor. Let D be a
floor diagram and δD its δ-gcd (the gcd between its edge weights and δ). Recall that we have
the division operator d
[
1
δ/δD
]
over the group algebra.
Definition 6.4. We define the correlated multiplicity of D to be
mδ(D) = d
[ 1
δ/δD
](∏
V
anV −1
V σδD(aV )
) ∏
e∈Eb(D)
we
∏
e∈E◦(D)
w2
e · (aλ0) ∈ Q[E],
where the first product is over the floors, the second product over the set of bounded edges
Eb(D), the last product over the set of possibly unbounded edges not adjacent to a flat vertex
E◦(D), and δλ0 = λ.
Figure 2. Floor diagrams.
We depict floor diagrams as on Figure 2, with edges oriented from bottom to top: sources
are at the bottom and sinks at the top. Floors are big-shaped with a bullet while flat vertices
are bullets on edges.
Example 6.3. We have two different floor diagrams on Figure 2. The first diagram has genus 3,
gcd 2 and tangency profile (2, 2,−2,−2). The second diagram has genus 4 and gcd 1, although
the gcd of the weights at infinity is 3. The tangency profile is (3, 3, 6,−3,−3,−6). For each
diagram, we can check that all vertices are balanced and the complement of all flat vertices is
a forest.
Floor diagrams encode some combinatorial types of curves in an expanded degeneration
of E × P1, which is the gluing of copies of E×P1 along their boundary divisors. As we prove in
Theorem 6.5, the latter are exactly the graphs appearing in the decomposition formula, up to
a redecoration.
� Sources and sinks correspond to marked points mapped to the boundary divisors of the
expanded degeneration.
� Flat vertices correspond to genus 0 components with an internal marked point mapped to
a fiber of E × P1.
� Floors correspond to genus 1 components in the class aV [E] + bV
[
P1
]
where bV is the flow
through V . Its intersection profile with the boundary divisor is prescribed by the adjacent
edges.
� Edges correspond to nodes in the domain mapped to the singular locus of the expanded
degeneration.
� Edges relating vertices which are not consecutive in the order on floors and flat vertices
should be subdivided. The new bivalent vertices would correspond to components mapped
to a fiber but without marked point.
Let w = (w1, . . . , wn) be a tangency profile and let δ be a common divisor. Let D be a floor
diagram and δD its δ-gcd (the gcd between its edge weights and δ). Recall that we have the
division operator d
[
1
δ/δD
]
over the group algebra.
40 T. Blomme and F. Carocci
Definition 6.4. We define the correlated multiplicity of D to be
mδ(D) = d
[
1
δ/δD
](∏
V
anV −1
V σδD(aV )
) ∏
e∈Eb(D)
we
∏
e∈E◦(D)
w2
e · (aλ0) ∈ Q[E],
where the first product is over the floors, the second product over the set of bounded edges Eb(D),
the last product over the set of possibly unbounded edges not adjacent to a flat vertex E◦(D),
and δλ0 = λ.
In the literature (for instance, [7, 13]), the flat vertices are not vertices but they are considered
as markings on the graph. Deleting flat vertices and merging the adjacent edges sharing the
same weight, the exponent of the weight we of a bounded edge (resp. end) in the multiplicity
would be 2 (resp. 1) if the edge carries a marking, and 3 (resp. 2) if it is not, matching the result
from [7].
This multiplicity mδ is a refinement of the multiplicity presented in [7], which is actually the
multiplicity m1.
6.2 Correspondence statement
We now use the degeneration formula to prove that the count of floor diagrams with the ad-
hoc multiplicity yields the correlated GW-invariant. We consider a degeneration Yt of P(O⊕L)
into n+ g − 1 components and one point constraint in each component. In particular, for a curve
f : C → Y0, among components of C mapped to a common component of Y0, exactly one may
contain a marked point.
Theorem 6.5. We have the following equality:〈〈
ptn+g−1, 1w1 , . . . , 1wn
〉〉δ
g,a[E],w
=
∑
D
mδ(D),
where we sum over floor diagrams D of genus g, in the class β, having tangency profile w.
The proof is standard application of the decomposition formula which we recall for sake of
completeness.
Up to the factor
∏
we
|Aut(Γ)| and the operator d
[
1
δ/δΓ
]
, the multiplicity of a degeneration graph Γ
given by the degeneration formula is as follows:∫
∏
V [[MV ]]δΓ
ev∗∆ ∪
n+g−1∏
1
ev∗i (pt).
This integral splits as a product over the vertices of the graph once we use the Künneth decom-
position of the diagonal. If (bi) is a basis and
(
b#i
)
the Poincaré dual basis, we have
∆ =
∑
i
(−1)rkbibi ⊗ b#i .
In our case, taking the basis (1, α, β,pt) of H•(E,Q), we have
∆ = 1⊗ pt + pt⊗ 1 + α⊗ β − β ⊗ α.
Expanding, it amounts to sum over all the possible insertions of Poincaré dual classes of E at
the extremities of bounded edges of Γ, which we call a Poincaré insertion. We now consider
a degeneration graph Γ and aim at transforming it into a floor diagram. This is the content of
Lemmas 6.6 and 6.7.
Correlated Gromov–Witten Invariants 41
Lemma 6.6. If Γ is a degeneration graph with non-zero multiplicity, we have the following:
(1) Inner vertices have genus 0 or 1, and a genus 1 vertex is adjacent to a marked point.
(2) A genus 0 vertices is bivalent and its class is of the form w
[
P1
]
.
(3) For a marked genus 0 vertex, adjacent Poincaré insertions are 1 ∈ H0(E,Q).
Proof. (1) Let V be a vertex with genus gV , valency nV , andmV = 0, 1 adjacent marked points.
The dimension of [[MV ]]
δΓ is nV +mV + gV − 1. For the multiplicity associated to a Poincaré
insertion to be non-zero, the sum of ranks of the insertions (Poincaré insertions coming from ∆
and the pt if mV = 1) needs to match this dimension. The sum of ranks of these insertions is
at most (nV − 1) + 2mV . The maximal rank of Poincaré insertions is nV − 1 and not nV due to
the existence of a relation between the position of points at infinity: choosing only pt insertions
at every flag yields 0 invariant. Therefore, we have the inequality
nV +mV + gV − 1 ⩽ nV − 1 + 2mV .
In other words, gV ⩽ mV , which implies (1): gV may only take the values 0, 1, and a genus 1
vertex is marked.
(2) Assume that gV = 0. As there all maps from a rational curve to an elliptic curve are
constant, the class associated to V is necessarily of the form w
[
P1
]
. Fibers varying in a 1-
dimensional family, we may only impose a unique point constraint:
� If mV = 1, we already have an interior point constraint, and the dimension is (nV +1)−1.
Thus, we have nV ⩽ 2.
� If mV = 0, the dimension is nV − 1, and we the sum of rank of the Poincaré insertions is
at most 1, so that we also have nV ⩽ 2.
In either case, as nV ⩾ 2 due to the class intersecting both divisors of the singular locus, we get
the bivalency statement from (2).
(3) Assume that V has genus 0 and is marked (mV = 1). Then the dimension of MV is 2,
and is already matched by the rank of the point insertion at the marked point, finishing the
proof. ■
Lemma 6.6 is a big step toward the transfiguration of a degeneration graph into a floor
diagram: edges are already weighted, vertices are split into flat bivalent vertices and floors as
advertised after Example 6.3. The total order comes from the total order on the components
of Y0. However, we still need three features:
� delete unmarked genus 0 vertices, which is achieved through Lemma 6.8,
� prove that the complement of genus 0 marked vertices is a forest with each component
containing a unique infinite vertex,
� the multiplicity matches mδ.
Consider Γ our degeneration graph. We can cut Γ at each genus 0 marked vertices, replacing
each bivalent vertex by two univalent vertices. The new graph (which may be disconnected) is
denoted by Γ̂. We then remove the infinite vertices and get a non-compact graph Γ̂◦ called open
cut graph.
Lemma 6.7. If Γ is a degeneration graph with non-zero multiplicity, and Γ̂◦ the associated open
cut graph, we have the following:
(1) Every connected component of Γ̂◦ has Euler characteristic 0.
42 T. Blomme and F. Carocci
(2) There are no compact connected component of Γ̂◦: every component of Γ̂ contains an
infinite vertex.
(3) Γ̂ has no cycle, and each connected component contains a unique infinite vertex.
(4) The Poincaré insertions are uniquely determined:
� for a marked genus 0 vertex, 1, 1 ∈ H•(E,Q),
� for an unmarked genus 0 vertex, 1,pt ∈ H•(E,Q),
� for a (marked) genus 1 vertex 1, pt, . . . ,pt ∈ H•(E,Q).
Proof. Let C be a connected component of the compact graph Γ̂. We denote by
� V1 the set of genus 1 (inner) vertices,
� V0 the set of genus 0 unmarked vertices,
� ∂ the set of univalent vertices resulting from the cut,
� ∂∞ the set of infinite vertices,
� Eb the set of edges not adjacent to a vertex of ∂ ∪ ∂∞.
Notice that we counting the flags adjacent to the vertices of V0 ∪ V1 in two ways, we have
that
∑
V1
nV + 2|V0| = 2|Eb|+ |∂|+ |∂∞|. The dimension of
∏
V ∈V0(C)∪V1(C)[[MV ]]
δΓ is∏
V ∈V0(C)∪V1(C)
[[MV ]]
δΓ =
∑
V1(C)
(nV + 1) +
∑
V0(C)
1
= 2|Eb|+ |∂|+ |∂∞|+ |V1| − |V0|.
For the invariant to be non-zero, it needs to match the sum of ranks of the insertions:
� the points for each vertex of ∂ through Lemma 6.6,
� the point insertions for every vertex of V1,
� the diagonal insertions for every edge of Eb.
Therefore, we have
2|Eb|+ |∂|+ |∂∞|+ |V1| − |V0| = |∂|+ 2|V1|+ |Eb|.
This yields |V1|+ |V0| − |Eb| − |∂∞| = 0. This quantity is precisely the Euler characteristic of C
where we deleted the vertices of ∂∞. Hence, we have (1).
As the Euler characteristic is 0, we have two possibilities:
� the connected component is compact with a unique cycle,
� the connected component has no cycle and contains a unique end, meaning |∂∞| = 1.
The point (2) forbids the first possibility so that we have (3). To prove (2), we may use [8,
Proposition 4.17] (and more precisely Step 5) that states that in the event where we have a cycle,
the invariant is 0 since we have no monodromy in this situation.
The statement about insertions follows from an induction on the trees of the forest Γ̂◦, pruning
the branches until we are left with the edges adjacent to ∂∞. ■
The local computation at the genus 1 vertices has already been carried out in Theorem 5.3.
We end with the local computation at the genus 0 vertices justifying the deletion of unmarked
genus 0 vertices before proving Theorem 6.5.
Correlated Gromov–Witten Invariants 43
Lemma 6.8. For bivalent vertices, we have the following invariants:
⟨⟨pt0, 1w, 1−w⟩⟩w0,0,(w,−w) = 1 · (0) and ⟨⟨ptw, 1−w⟩⟩w0,0,(w,−w) =
1
w
· (0).
Proof. Curves in the class w
[
P1
]
are mapped to some fiber of E × P1. Therefore, the only
correlated invariant which is non-zero in this situation is for (0). The uncorrelated computation
has already been carried out in [11, Lemma 3.3]: in both situations there is a unique map to
the fiber, but with a Z/wZ automorphism group in the second case. ■
Proof of Theorem 6.5. The number of marked points is equal to n + g − 1. We consider
a degeneration Yt of E × P1 with a central fiber Y0 having n + g − 1 irreducible components.
We apply the degeneration formula from Theorem 4.12, putting exactly one point constraint
per component of the central fiber. Degeneration formula asserts that the correlated invariant
we care about is a sum over the degeneration graphs. We need to show that the degeneration
graphs with non-zero multiplicity are actually the floor diagrams involved in the sum, and that
they share the same multiplicity.
Given a degeneration graph Γ, Lemma 6.6 ensures that the vertices have the right genus and
valency while Lemma 6.7 (3) states that the complement of all flat vertices has no cycle and
each component has a unique end, so that Γ is indeed a floor diagram. In particular, there are
no automorphisms.
Furthermore, by Lemma 6.7 (4), the Poincaré insertions coming from the Künneth decompo-
sition of the diagonal are uniquely determined, so that the multiplicity splits as a product over
the vertices. The multiplicity at genus 0 vertices is provided by Lemma 6.8 and tells us that we
may forget about genus 0 unmarked vertices, as their multiplicity cancels with a factor of
∏
ewe
when merging the adjacent edges.
The local multiplicity at the genus 1 vertices is provided by Theorem 5.3, yielding the mul-
tiplicity mδ. ■
Remark 6.9. The diagram setting can be adapted to deal with invariants including boundary
constraints. To simplify notations, we only consider the case of interior point constraints.
6.3 Quasi-modularity
We now use the floor diagrams to prove some regularity statements for the invariants, starting
with the generating series in the base direction: varying a in the curve class a[E] + b
[
P1
]
. For
that purpose, we assume that L = O, so that we may choose λ0 = 0 and the term (aλ0) in the
diagram multiplicities disappears.
We consider quasi-modular forms with values in the vector space Q[Torδ(E)]. Usual quasi-
modular definitions extend naturally to the setting of functions in a vector space.
Definition 6.10. Let V be a complex vector space and f : H → V a meromorphic function on
the Poincaré half-plane and Γ ⊂ SL2(Z) a subgroup.
(1) We say that f is modular of weight k if for any
(
a b
c d
)
∈ Γ we have
(cτ + d)kf
(
aτ + b
cτ + d
)
= f(τ).
(2) We say that f is quasi-modular of weight k and depth s if there exists functions fj : H → V
such that for any
(
a b
c d
)
∈ Γ we have
(cτ + d)kf
(
aτ + b
cτ + d
)
=
s∑
0
fj(τ)
(
c
cτ + d
)j
.
44 T. Blomme and F. Carocci
A function f : H → V is (quasi-)modular if and only if all its coordinate functions are
(quasi-)modular. Furthermore, in the quasi-modularity case, all fj are also quasi-modular forms
and f0 = f . Furthermore, if the vector space V is endowed with an C-algebra structure, the
product of quasi-modular forms is also quasi-modular. If ( 1 1
0 1 ) ∈ Γ, we can set q = e2iπτ and
develop the quasi-modular forms in Fourier series, or conversely consider generating series in q
as meromorphic functions potentially quasi-modular.
Example 6.11. We already saw that the generating series
∑∞
a=1 σδ(a)q
a are quasi-modular
forms for the congruence subgroup Γ0(δ). Its derivatives
∑∞
a=1 a
nσδ(a)q
a are as well.
Theorem 6.12. Let w be a tangency profile of length n. The generating series
∞∑
a=1
〈〈
ptn+g−1
〉〉δ
g,a[E],w
qa
is a quasi-modular form for the congruence subgroup Γ0(δ) with values in the group algebra
C[Torδ(E)].
Proof. Up to the labeling of floors by (aV ), there is a finite number of floor diagrams. It thus
suffices to prove the statement for each unlabelled floor diagram.
Let D a floor diagram without floor labels and let W =
∏
Eb(D)we
∏
E◦(D)w
2
e . Let D
(
(aV )
)
be the floor diagram obtained by labeling the vertices with (aV ). As d
[
1
δ/δΓ
]
is compatible with
the product, we have the following factorization
∞∑
a=1
( ∑
ΣaV =a
mδ(D((aV )))
)
qa =
∑
(aV )
mδ(D
(
(aV )
)
)q
∑
aV
=W
∑
(aV )
d
[
1
δ/δD
](∏
V
anV −1
V σδD(aV )q
aV
)
=W · d
[
1
δ/δD
]∏
V
( ∞∑
aV =1
anV −1
V σδD(aV )q
aV
)
.
As each series is a quasi-modular form, so is their product. ■
Remark 6.13. The optimality of δ in the statement is ensured by the existence of floor diagrams
with gcd δ.
Remark 6.14. If we do not assume that L = O anymore, the coefficients of the generating
series are multiplied by (aλ0), where δλ0 = λ. The naive version of quasi-modularity is not
satisfied anymore. It is not really clear which kind of regularity to expect for these functions,
with values in C[E] and not the finite-dimensional C[Torδ(E)].
6.4 Piecewise polynomiality
We now vary the tangency orders, considering them as variables. We study the function
N δ
a,g : w = (w1, . . . , wn) 7−→ ⟨⟨ptn+g−1, 1w1 , . . . , 1wn⟩⟩δg,a[E],w ∈ Q[Torδ(E)].
For this definition to make sense, we need to restrict to the sublattice where all the wi are
divisible by δ, so that the δ-refinement makes sense.
Up to the labeling of edges by their weight, there is a finite number of floor diagrams of genus g
with n ends and with a class of the form a[E] + ∗
[
P1
]
. The orientation is still part of the data,
only the edge weights are missing. Let D be one of them. To make D into a true floor diagram,
Correlated Gromov–Witten Invariants 45
one only needs to add weights to the edges. We denote by ΩD the set of weightings ω of D, i.e.,
functions from edges to positive integers that make the floors and flat vertices balanced. For
a weighting ω ∈ ΩD, we denote by δω the gcd between δ and its coordinates. The tangency profile
is the restriction of the weighting to the infinite ends. For a fixed tangency profile w, we denote
by ΩD(w) the set of weightings that induce w on the ends of D. Although by assumption w is
divisible by δ, ω may not be.
The multiplicity from Definition 6.4 contains as a factor a monomial in the edge weights,
which we denote by fD,
fD : ω ∈ ΩD 7−→
∏
Eb(D)
we
∏
E◦(D)
we,
where the first product is over bounded edges of D and the second product is over edges not
adjacent to a flat vertex. The remaining part of the multiplicity depends on the floors weights aV
but also on the weighting ω through its gcd δω. We denote it by
ΦD(δω) = d
[
1
δ/δω
](∏
V
anV −1
V σδω(aV )
)
,
so that mδ
(
D(ω)
)
= ΦD(δω)fD(ω). Even if we only care about the weighting that make the
vertices balanced, this function is defined for all assignations of weights to the edges, not only
the elements of ΩD. Before going to the main statement of the section, we reformulate the
expression of ΦD(δω) to suit our purpose.
Lemma 6.15. For a chosen diagram D up to edge weights, there exists functions ΥD
d such that
ΦD(δω) =
∑
d|δω
ΥD
d (δω)ϑδ/d.
Proof. We use the expression of σδω highlighting the role of the ϑd
σδω(aV ) =
∑
d|δω
Υδω
d (aV )ϑδω/d.
As the map d
[
1
δ/δω
]
is compatible with the product and d
[
1
δ/δω
]
(ϑδω/d) = ϑδ/d, we have that
ΦD(δω) =
∏
V
(
anV −1
V
∑
dV |δω
Υδω
dV
(aV )ϑδ/dV
)
.
We expand the product and use that ϑδ/d1ϑδ/d2 = ϑδ/ gcd(d1,d2),
ΦD(δω) =
(∏
anV −1
V
)∑
d|δω
( ∑
dV |δω
gcd(δV )=d
∏
V
Υδω
dV
(aV )
)
ϑδ/d.
We thus have the existence and an expression for the desired coefficients ΥD
d (δω), which only
depend on δω and d once D is fixed. ■
Our statement relies on [3, Theorem 4.2] which is a generalization of [45, Theorem 1]. To
apply the theorem, we provide a second description of the weighting set ΩD(w) as the set of
lattice points of a flow polytope. Let AD be the adjacency matrix of D: rows are indexed by
46 T. Blomme and F. Carocci
vertices (floors, flat vertices, sinks, sources) and columns by edges. Coefficients are given by the
following rule:
AD(V, e) =
0 if V /∈ e,
1 if e ends at V,
−1 if e starts at V.
In particular, for an element ω ∈ Zn+|Eb(D)|, the image ADω is exactly the divergence at each
vertex: the difference between incoming and outgoing weights at V . Therefore, if d = (dV ) is
the vector with coordinate |wi| for an infinite vertex and 0 else, we have
ΩD(w) =
{
ω ∈ Nn+Eb(D) s.t. ADω = d
}
.
Theorem 6.16. For fixed a, g, n, δ, the function w 7→ N δ
a,g(w) is piecewise polynomial in the
sense that there exists piecewise polynomial functions Pd(w) for d|δ such that
N δ
a,g(w) =
∑
d|δ
Pd(w)ϑd.
Proof. Up to the weighting of the edges, there is a finite number of floor diagrams of genus g
with class of the form a[E] + ∗
[
P1
]
and n ends. Grouping them together, we get
N δ
a,g(w) =
∑
D
N δ
D(w), with N δ
D(w) =
∑
ω∈ΩD(w)
ΦD(δω)fD(ω).
Thus, to prove piecewise polynomiality, it is sufficient to consider a unique unweighed floor
diagram D.
Let D be such an unweighed floor diagram. If there was no ΦD(δω), [3, Theorem 4.2] already
applies and yields piecewise polynomiality. Due to the presence of this factor depending on the
gcd δω, our second step is thus to first use the expression of ΦD(δω), split the sum and perform
a multiplicative summation by parts (as described before Theorem 5.3). We aim to change the
sum over ω and a given δω into a sum over the ω in a sublattice. Using Lemma 6.15, we have
N δ
D(w) =
∑
ω∈ΩD(w)
(∑
d|δω
ΥD
d (δω)ϑδ/d
)
fD(ω).
To perform the multiplicative summation by parts, we can find coefficients γD
d , which are linear
combinations of the ϑδ/d, such that for each δω we have∑
d|δω
γD
d =
∑
d|δω
ΥD
d (δω)ϑδ/d.
Such coefficients exist because the equations given for each δω form a triangular linear system.
We can now switch the two sums and get rid of the dependence in the gcd δω
N δ
D(w) =
∑
ω∈ΩD(w)
(∑
d|δω
γD
d
)
fD(ω) =
∑
d|δ
γD
d
∑
ω∈ΩD(w)
d|δω
fD(ω).
The last step is now to apply [3, Theorem 4.2]. For each d|δ, we can rewrite∑
ω∈ΩD(w)
d|ω
fD(ω) =
∑
ω′∈ΩD(w/d)
fD(dω
′).
Correlated Gromov–Witten Invariants 47
The function is piecewise quasi-polynomial with respect to the chamber complex of AD. Fur-
thermore, as AD is actually unimodular (it is a submatrix of the root system An−1, see [3,
Example 4.4]), we in fact get piecewise polynomiality and not only quasi-polynomiality. Sum-
ming over all diagrams, we conclude. ■
Example 6.17. Assume δ = p is a prime number and D is a diagram up to edge weights. Then
we have two kinds of weightings:
� the weightings with gcd 1 and multiplicity ΥD
1 (1)ϑp · fD(ω),
� the weightings with gcd p and multiplicity
(
ΥD
1 (p)ϑp +ΥD
p (p)ϑ1
)
· fD(ω).
The multiplicative summation by parts tells us to count all weightings with the multiplicity of
gcd 1 weightings, and correct by counting the gcd p weightings with multiplicity((
ΥD
1 (p)−ΥD
1 (1)
)
ϑp +ΥD
p (p)ϑ1
)
fD(ω).CORRELATED GROMOV-WITTEN INVARIANTS 49
•
•
•
•w
w
w
w
•
•
•
w
•w
w
w
•
•
•
w
w
•w
w
•
•
•
w
w
w
•w
•
•
•w
w
•w1 w2
•
•
w
•w
•w1 w2
Figure 3. Floor diagrams of genus 3.
Example 6.18. We end this section with an explicit computation. We take n = 2 and g = 3
and a chosen δ, for instance δ = 2. As n = 2, the tangency profile is necessarily (w,−w) and
w is divisible by δ. Up to weights of the edges and floors, we only have six floor diagrams,
of two different kinds. For the first four floor diagrams, the multiplicity is a constant term in
Q[Torδ(E)] times w9, which is indeed a polynomial in w. We now care about the diagrams
of the second kind. To choose a weighting of the edges, one needs to choose a splitting
w = w1 + w2. If they are both even, the gcd δω of the diagram is 2, otherwise both of them
are odd and the gcd is 1. The multiplicity is
a21a
2
2d
[ 1
2/δω
]
(σδω(a1)σδω(a2))w3w2
1w
3
2,
where we have the two expressions
σ1(a) =σ(a)ϑ1,
σ2(a) =σ2(a)ϑ1 + (σ(a)− σ2(a))ϑ2.
Therefore, in the odd case the multiplicity is
a21a
2
2σ(a1)σ(a2)ϑ2 · w3w2
1w
3
2,
and in the even case
a21a
2
2
(
σ2(a1)σ2(a2)ϑ1 +
(
σ(a1)σ(a2)− σ2(a1)σ2(a2)
)
ϑ2
)
· w3w2
1w
3
2.
To get the result, we sum the multiplicity in the odd case over 1 ⩽ w1 ⩽ w − 1 to get a first
polynomial, and sum the difference between the two multiplicity over the even w1 satisfying
the same inequalities. ♢
References
[AB17] Federico Ardila and Erwan Brugallé. The double Gromov–Witten invariants of Hirzebruch surfaces
are piecewise polynomial. International Mathematics Research Notices, 2017(2):614–641, 2017.
[AC14] Dan Abramovich and Qile Chen. Stable logarithmic maps to deligne-faltings pairs ii. Asian Journal
of Mathematics, 18(3):465–488, 2014.
Figure 3. Floor diagrams of genus 3.
Example 6.18. We end this section with an explicit computation. We take n = 2 and g = 3
and a chosen δ, for instance δ = 2. As n = 2, the tangency profile is necessarily (w,−w) and w
is divisible by δ. Up to weights of the edges and floors, we only have six floor diagrams, of two
different kinds. For the first four floor diagrams, the multiplicity is a constant term in Q[Torδ(E)]
times w9, which is indeed a polynomial in w. We now care about the diagrams of the second
kind. To choose a weighting of the edges, one needs to choose a splitting w = w1 + w2. If they
are both even, the gcd δω of the diagram is 2, otherwise both of them are odd and the gcd is 1.
The multiplicity is
a21a
2
2d
[
1
2/δω
]
(σδω(a1)σδω(a2))w
3w2
1w
3
2,
where we have the two expressions
σ1(a) = σ(a)ϑ1, σ2(a) = σ2(a)ϑ1 + (σ(a)− σ2(a))ϑ2.
Therefore, in the odd case the multiplicity is a21a
2
2σ(a1)σ(a2)ϑ2 · w3w2
1w
3
2, and in the even case
a21a
2
2
(
σ2(a1)σ
2(a2)ϑ1 +
(
σ(a1)σ(a2)− σ2(a1)σ
2(a2)
)
ϑ2
)
· w3w2
1w
3
2.
To get the result, we sum the multiplicity in the odd case over 1 ⩽ w1 ⩽ w − 1 to get a first
polynomial, and sum the difference between the two multiplicity over the even w1 satisfying the
same inequalities.
48 T. Blomme and F. Carocci
Acknowledgements
T.B. is supported by the SNF grant 204125; F.C. is supported by the Ambizione grant PZ00P2-
208699/1. F.C. is also partially supported by the MIUR Excellence Department Project Mat-
Mod@TOV, CUP E83C23000330006, awarded to the Department of Mathematics, University of
Rome Tor Vergata, and also acknowledges the support of the PRIN Project “Moduli spaces and
birational geometry” 2022L34E7W. The authors would like to thank D. Ranganathan, A. Ku-
maran, S. Molcho for helpful discussion around the subject of this paper.
References
[1] Abramovich D., Chen Q., Stable logarithmic maps to Deligne–Faltings pairs II, Asian J. Math. 18 (2014),
465–488, arXiv:1102.4531.
[2] Abramovich D., Marcus S., Wise J., Comparison theorems for Gromov–Witten invariants of smooth pairs
and of degenerations, Ann. Inst. Fourier (Grenoble) 64 (2014), 1611–1667, arXiv:1207.2085.
[3] Ardila F., Brugallé E., The double Gromov–Witten invariants of Hirzebruch surfaces are piecewise polyno-
mial, Int. Math. Res. Not. 2017 (2017), 614–641, arXiv:1412.4563.
[4] Behrend K., Fantechi B., The intrinsic normal cone, Invent. Math. 128 (1997), 45–88, arXiv:alg-
geom/9601010.
[5] Birkenhake C., Lange H., Complex abelian varieties, 2nd ed., Grundlehren Math. Wiss., Vol. 302, Springer,
Berlin, 2004.
[6] Blomme T., Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas,
arXiv:2205.07684.
[7] Blomme T., Floor diagrams and enumerative invariants of line bundles over an elliptic curve, Compos. Math.
159 (2023), 1741–1790, arXiv:2112.05439.
[8] Blomme T., Gromov–Witten invariants of bielliptic surfaces, J. Lond. Math. Soc. 111 (2025), e70081,
61 pages, arXiv:2401.01627.
[9] Borne N., Vistoli A., Parabolic sheaves on logarithmic schemes, Adv. Math. 231 (2012), 1327–1363,
arXiv:1001.0466.
[10] Bosch S., Lütkebohmert W., Raynaud M., Néron models, Ergeb. Math. Grenzgeb., Vol. 21, Springer, Berlin,
1990.
[11] Bousseau P., Refined floor diagrams from higher genera and lambda classes, Selecta Math. 27 (2021), 43,
42 pages, arXiv:1904.10311.
[12] Brugallé E., Mikhalkin G., Enumeration of curves via floor diagrams, C. R. Math. Acad. Sci. Paris 345
(2007), 329–334, arXiv:0706.0083.
[13] Brugallé E., Mikhalkin G., Floor decompositions of tropical curves: the planar case, in Proceedings of
Gökova Geometry–Topology Conference 2008, Gökova Geometry/Topology Conference (GGT), Gökova,
2009, 64–90, arXiv:0812.3354.
[14] Carocci F., Nabijou N., Rubber tori in the boundary of expanded stable maps, J. Lond. Math. Soc. 109
(2024), e12874, 36 pages, arXiv:2109.07512.
[15] Chen Q., The degeneration formula for logarithmic expanded degenerations, J. Algebraic Geom. 23 (2014),
341–392, arXiv:1009.4378.
[16] Esteves E., Gagné M., Kleiman S., Autoduality of the compactified Jacobian, J. London Math. Soc. 65
(2002), 591–610, arXiv:math.AG/9911071.
[17] Fantechi B., Göttsche L., Illusie L., Kleiman S.L., Nitsure N., Vistoli A., Fundamental algebraic geome-
try: Grothendieck’s FGA explained, Math. Surveys Monogr., Vol. 123, American Mathematical Society,
Providence, RI, 2005.
[18] Fulton W., Intersection theory, Ergeb. Math. Grenzgeb., Vol. 2, Springer, Berlin, 1984.
[19] Gross M., Siebert B., Logarithmic Gromov–Witten invariants, J. Amer. Math. Soc. 26 (2013), 451–510,
arXiv:1102.4322.
[20] Grothendieck A., Technique de descente et théorèmes d’existence en géométrie algébrique. VI. Les schémas
de Picard: propriétés générales, in Séminaire Bourbaki, Vol. 7, Société Mathématique de France, Paris,
1995, Exp. No. 236, 221–243.
https://doi.org/10.4310/AJM.2014.v18.n3.a5
http://arxiv.org/abs/1102.4531
https://doi.org/10.5802/aif.2892
http://arxiv.org/abs/1207.2085
https://doi.org/10.1093/imrn/rnv379
http://arxiv.org/abs/1412.4563
https://doi.org/10.1007/s002220050136
http://arxiv.org/abs/alg-geom/9601010
http://arxiv.org/abs/alg-geom/9601010
https://doi.org/10.1007/978-3-662-06307-1
http://arxiv.org/abs/2205.07684
https://doi.org/10.1112/s0010437x23007285
http://arxiv.org/abs/2112.05439
https://doi.org/10.1112/jlms.70081
http://arxiv.org/abs/2401.01627
https://doi.org/10.1016/j.aim.2012.06.015
http://arxiv.org/abs/1001.0466
https://doi.org/10.1007/978-3-642-51438-8
https://doi.org/10.1007/s00029-021-00667-w
http://arxiv.org/abs/1904.10311
https://doi.org/10.1016/j.crma.2007.07.026
http://arxiv.org/abs/0706.0083
http://arxiv.org/abs/0812.3354
https://doi.org/10.1112/jlms.12874
http://arxiv.org/abs/2109.07512
https://doi.org/10.1090/S1056-3911-2013-00614-1
http://arxiv.org/abs/1009.4378
https://doi.org/10.1112/S002461070100309X
http://arxiv.org/abs/math.AG/9911071
https://doi.org/10.1090/surv/123
https://doi.org/10.1007/978-3-662-02421-8
https://doi.org/10.1090/S0894-0347-2012-00757-7
http://arxiv.org/abs/1102.4322
Correlated Gromov–Witten Invariants 49
[21] Janda F., Pandharipande R., Pixton A., Zvonkine D., Double ramification cycles with target varieties,
J. Topol. 13 (2020), 1725–1766, arXiv:1812.10136.
[22] Kajiwara T., Kato K., Nakayama C., Logarithmic abelian varieties, Nagoya Math. J. 189 (2008), 63–138.
[23] Kato F., Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), 215–232.
[24] Kim B., Logarithmic stable maps, in New Developments in Algebraic Geometry, Integrable Systems and
Mirror Symmetry (RIMS, Kyoto, 2008), Adv. Stud. Pure Math., Vol. 59, Mathematical Society of Japan,
Tokyo, 2010, 167–200, arXiv:0807.3611.
[25] Kim B., Lho H., Ruddat H., The degeneration formula for stable log maps, Manuscripta Math. 170 (2023),
63–107, arXiv:1803.04210.
[26] Li J., Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001),
509–578.
[27] Li J., A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), 199–293,
arXiv:math.AG/0110113.
[28] Mandel T., Ruddat H., Descendant log Gromov–Witten invariants for toric varieties and tropical curves,
Trans. Amer. Math. Soc. 373 (2020), 1109–1152, arXiv:1612.02402.
[29] Manolache C., Virtual pull-backs, J. Algebraic Geom. 21 (2012), 201–245, arXiv:0805.2065.
[30] Marcus S., Wise J., Logarithmic compactification of the Abel–Jacobi section, Proc. Lond. Math. Soc. 121
(2020), 1207–1250, arXiv:1708.04471.
[31] Maulik D., Pandharipande R., Thomas R.P., Curves on K3 surfaces and modular forms, J. Topol. 3 (2010),
937–996, arXiv:1001.2719.
[32] Maulik D., Ranganathan D., Logarithmic enumerative geometry for curves and sheaves, Camb. J. Math. 13
(2025), 51–172, arXiv:2311.14150.
[33] Melo M., Compactifications of the universal Jacobian over curves with marked points, arXiv:1509.06177.
[34] Melo M., Rapagnetta A., Viviani F., Fourier–Mukai and autoduality for compactified Jacobians. I, J. Reine
Angew. Math. 755 (2019), 1–65, arXiv:1207.7233.
[35] Milne J.S., Abelian varieties (v2.00), 2008, available at https://www.jmilne.org/math/.
[36] Mochizuki S., Topics in absolute anabelian geometry I: generalities, J. Math. Sci. Univ. Tokyo 19 (2012),
139–242.
[37] Molcho S., Ulirsch M., Wise J., The logarithmic Deligne pairing, Private communication.
[38] Molcho S., Wise J., The logarithmic Picard group and its tropicalization, Compos. Math. 158 (2022), 1477–
1562, arXiv:1807.11364.
[39] Mumford D., Ramanujam C.P., Manin J.I., Abelian varieties, Stud. Math., Vol. 5, Oxford University Press,
1974.
[40] Oberdieck G., Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties,
Forum Math. Sigma 10 (2022), e21, 46 pages, arXiv:2102.11622.
[41] Ogus A., Lectures on logarithmic algebraic geometry, Cambridge Stud. Adv. Math., Vol. 178, Cambridge
University Press, Cambridge, 2018.
[42] Ranganathan D., Kumaran A.U., Logarithmic Gromov–Witten theory and double ramification cycles,
J. Reine Angew. Math. 809 (2024), 1–40, arXiv:2212.11171.
[43] Ranganathan D., Santos-Parker K., Wise J., Moduli of stable maps in genus one and logarithmic geometry, II,
Algebra Number Theory 13 (2019), 1765–1805, arXiv:1709.00490.
[44] Ranganathan D., Wise J., Rational curves in the logarithmic multiplicative group, Proc. Amer. Math. Soc.
148 (2020), 103–110, arXiv:1901.08489.
[45] Sturmfels B., On vector partition functions, J. Combin. Theory Ser. A 72 (1995), 302–309.
https://doi.org/10.1112/topo.12174
http://arxiv.org/abs/1812.10136
https://doi.org/10.1017/S002776300000951X
https://doi.org/10.1142/S0129167X0000012X
https://doi.org/10.2969/aspm/05910167
http://arxiv.org/abs/0807.3611
https://doi.org/10.1007/s00229-021-01361-z
http://arxiv.org/abs/1803.04210
https://doi.org/10.4310/jdg/1090348132
https://doi.org/10.4310/jdg/1090351102
http://arxiv.org/abs/math.AG/0110113
https://doi.org/10.1090/tran/7936
http://arxiv.org/abs/1612.02402
https://doi.org/10.1090/S1056-3911-2011-00606-1
http://arxiv.org/abs/0805.2065
https://doi.org/10.1112/plms.12365
http://arxiv.org/abs/1708.04471
https://doi.org/10.1112/jtopol/jtq030
http://arxiv.org/abs/1001.2719
https://doi.org/10.4310/cjm.250319031722
http://arxiv.org/abs/2311.14150
http://arxiv.org/abs/1509.06177
https://doi.org/10.1515/crelle-2017-0009
https://doi.org/10.1515/crelle-2017-0009
http://arxiv.org/abs/1207.7233
https://www.jmilne.org/math/
https://doi.org/10.1112/s0010437x22007527
http://arxiv.org/abs/1807.11364
https://doi.org/10.1017/fms.2022.10
http://arxiv.org/abs/2102.11622
https://doi.org/10.1017/9781316941614
https://doi.org/10.1017/9781316941614
https://doi.org/10.1515/crelle-2023-0100
http://arxiv.org/abs/2212.11171
https://doi.org/10.2140/ant.2019.13.1765
http://arxiv.org/abs/1709.00490
https://doi.org/10.1090/proc/14749
http://arxiv.org/abs/1901.08489
https://doi.org/10.1016/0097-3165(95)90067-5
1 Introduction
1.1 Setting
1.2 The Albanese evaluation
1.2.1 Case of X times P^1
1.2.2 Case of a non-trivial bundle
1.3 Correlated classes and correlated GW invariants
1.4 Properties of the correlated classes
1.4.1 Deformation invariance
1.4.2 Degeneration formula
1.5 The elliptic case
1.5.1 Local invariants
1.5.2 Floor diagrams and regularity
1.6 Future directions
2 Logarithmic curves, line bundles and stable maps
2.1 Curves and line bundles
2.1.1 Logarithmic curves
2.1.2 Tropical curves
2.1.3 Piecewise linear functions
2.1.4 Line bundles
2.2 Logarithmic line bundles and trivializations
2.3 Logarithmic stable maps to P^1-bundles and their degenerations
3 Correlated virtual class
3.1 Recollection on Albanese varieties
3.1.1 Complex setting
3.1.2 Algebraic definition
3.1.3 Self-duality for smooth curves
3.2 Refinement of the moduli spaces
3.2.1 Morphism to Alb(X)
3.2.2 Refinement by correlation
3.2.3 Correlated Gromov–Witten invariants
3.3 Deformation invariants and relations of correlated classes
4 Refined decomposition formula
4.1 Recollection on the degeneration formula
4.2 Toward the refined degeneration formula
4.3 General degeneration formula
4.4 Degeneration in the elliptic case
5 Computation of local invariants
5.1 General considerations and statement
5.1.1 Uncorrelated case
5.1.2 Statement
5.1.3 Applications
5.2 Contribution of a fixed cover
5.3 Computation of the (theta_0)-coefficient
5.4 Case of other correlators
5.4.1 Dependence on the order
5.4.2 Coefficients, induction relation and multiplicativity
5.4.3 Computation for powers of primes
6 Floor diagrams and regularity
6.1 Floor diagrams and their multiplicities
6.2 Correspondence statement
6.3 Quasi-modularity
6.4 Piecewise polynomiality
References
|
| id | nasplib_isofts_kiev_ua-123456789-213530 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T20:39:16Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Blomme, Thomas Carocci, Francesca 2026-02-18T11:26:10Z 2025 Correlated Gromov-Witten Invariants. Thomas Blomme and Francesca Carocci. SIGMA 21 (2025), 046, 49 pages 1815-0659 2020 Mathematics Subject Classification: 14N35; 14N10; 14J26 arXiv:2409.09472 https://nasplib.isofts.kiev.ua/handle/123456789/213530 https://doi.org/10.3842/SIGMA.2025.046 We introduce a geometric refinement of Gromov-Witten invariants for ℙ¹-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a refinement of the degeneration formula, keeping track of the correlation. Finally, combining certain invariance properties of the correlated invariant, a local computation, and the refined degeneration formula, we follow floor diagram techniques to prove regularity results for the generating series of the invariants in the case of ℙ¹-bundles over elliptic curves. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces. T.B. is supported by the SNF grant 204125; F.C. is supported by the Ambizione grant PZ00P2208699/1. F.C. is also partially supported by the MIUR Excellence Department Project MatMod@TOV, CUPE83C23000330006, awarded to the Department of Mathematics, University of Rome Tor Vergata, and also acknowledges the support of the PRIN Project “Moduli spaces and birational geometry” 2022L34E7W. The authors would like to thank D. Ranganathan, A. Kumaran, and S. Molcho for helpful discussions around the subject of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Correlated Gromov-Witten Invariants Article published earlier |
| spellingShingle | Correlated Gromov-Witten Invariants Blomme, Thomas Carocci, Francesca |
| title | Correlated Gromov-Witten Invariants |
| title_full | Correlated Gromov-Witten Invariants |
| title_fullStr | Correlated Gromov-Witten Invariants |
| title_full_unstemmed | Correlated Gromov-Witten Invariants |
| title_short | Correlated Gromov-Witten Invariants |
| title_sort | correlated gromov-witten invariants |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213530 |
| work_keys_str_mv | AT blommethomas correlatedgromovwitteninvariants AT caroccifrancesca correlatedgromovwitteninvariants |