Tau Function and Moduli of Meromorphic Quadratic Differentials
The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most 𝑛 simple poles on genus 𝑔 complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
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Інститут математики НАН України
2022
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| Цитувати: | Tau Function and Moduli of Meromorphic Quadratic Differentials. Dmitry Korotkin and Peter Zograf. SIGMA 18 (2022), 001, 10 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859592742172098560 |
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| author | Korotkin, Dmitry Zograf, Peter |
| author_facet | Korotkin, Dmitry Zograf, Peter |
| citation_txt | Tau Function and Moduli of Meromorphic Quadratic Differentials. Dmitry Korotkin and Peter Zograf. SIGMA 18 (2022), 001, 10 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most 𝑛 simple poles on genus 𝑔 complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.
|
| first_indexed | 2026-03-13T23:50:37Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 001, 10 pages
Tau Function and Moduli
of Meromorphic Quadratic Differentials
Dmitry KOROTKIN ab and Peter ZOGRAF bc
a) Department of Mathematics and Statistics, Concordia University,
1455 de Maisonneuve West, Montreal, H3G 1M8 Quebec, Canada
E-mail: dmitry.korotkin@concordia.ca
b) Euler International Mathematical Institute, Pesochnaja nab. 10,
Saint Petersburg, 197022 Russia
E-mail: zograf@pdmi.ras.ru
c) Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O. 29,
Saint Petersburg, 199178 Russia
Received August 09, 2021, in final form December 28, 2021; Published online January 03, 2022
https://doi.org/10.3842/SIGMA.2022.001
Abstract. The Bergman tau functions are applied to the study of the Picard group of
moduli spaces of quadratic differentials with at most n simple poles on genus g complex
algebraic curves. This generalizes our previous results on moduli spaces of holomorphic
quadratic differentials.
Key words: quadratic differentials; tau function; moduli spaces
2020 Mathematics Subject Classification: 14H15; 14H70; 14K20; 30F30
To Leon Takhtajan on occasion
of his 70 th birthday
1 Introduction and statement of results
The theory of the Bergman tau function was applied to the study of geometry of various moduli
spaces in [3, 7, 9, 10, 11]; see [8] for a review of known results.
The present paper is a continuation of [11]. Here we use Bergman tau functions to study
the geometry of the moduli space Qg,n of quadratic differentials with n simple poles on genus g
complex algebraic curves. More precisely, the space Qg,n is defined as the set of isomorphism
classes of pairs (C, q), where C is a smooth genus g complex curve with n labeled distinct
marked points, and q is a meromorphic quadratic differential on C with at most simple poles at
the marked points and no other poles (throughout the paper we will assume that 2g+n > 3). It
is well known that Qg,n is naturally isomorphic to T ∗Mg,n, the total space of the holomorphic
cotangent bundle on the moduli space Mg,n of pointed complex algebraic curves.
The bundle T ∗Mg,n can be extended to the Deligne–Mumford boundary of Mg,n in two
natural ways: first, as the cotangent bundle T ∗Mg,n on Mg,n, the Deligne–Mumford moduli
space of stable curves, and second, as the total space of the direct image π∗ω
2
g,n, where ωg,n is
the relative dualizing sheaf on the universal curve π : Cg,n → Mg,n. These two extensions are
rather close to each other: namely, detT ∗Mg,n = detπ∗ω
2
g,n − δDM, where δDM = Mg,n \Mg,n
This paper is a contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quan-
tum in honor of Leon Takhtajan.
The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html
mailto:dmitry.korotkin@concordia.ca
zograf@pdmi.ras.ru
https://doi.org/10.3842/SIGMA.2022.001
https://www.emis.de/journals/SIGMA/Takhtajan.html
2 D. Korotkin and P. Zograf
is the Deligne–Mumford boundary class (a detailed analytic treatment of these subjects can be
found in [13]). In this paper we will use the second extension and put Qg,n = π∗ω
2
g,n.
The moduli space Qg,n has an open dense subset Q
(
14g−4+n,−1n
)
that consists of isomor-
phism classes of pairs (C, q), where C is a smooth curve, and q has exactly 4g − 4 + n simple
zeros and n simple poles. We call Q
(
14g−4+n,−1n
)
the principal stratum.
Denote by PQg,n = Qg,n/C∗ the projectivization of Qg,n, where C∗ acts on quadratic differen-
tials by multiplication. The complement PQg,n\PQ
(
14g−4+n,−1n
)
is a union of divisors that we
denote by D0
deg, D
∞
deg, and DDM (the subscript deg stands for degenerate, as opposed to differen-
tials in Q
(
14g−4+n,−1n
)
that we call generic). The divisor D0
deg is the closure of the set of differ-
entials q with multiple zeros. The divisor D∞
deg is the closure of n strata PQ
(
14g−5+n, 0,−1n−1
)
(one for each pole) parameterizing differentials with 4g − 5 + n simple zeros, n− 1 simple poles
and one marked ordinary point (where a zero and a pole of a generic differential coalesce). Fi-
nally, DDM is the pullback to PQg,n of the Deligne–Mumford boundary of Mg,n. The latter
consists of the divisor Dirr of irreducible nodal curves with n marked points, and divisors Dj,k
parameterizing reducible curves with components of genera j and g − j having k and n − k
marked points respectively, where j = 0, . . . , [g/2], k = 0, . . . , n and 2 < 2j + k < 2g + n − 2.
(Note that while a quadratic differential may have only simple poles at the n marked points, it
may have poles up to order n at the nodes.)
Denote by L → PQg,n the tautological line bundle associated with the projectivization
Qg,n → PQg,n, and put ϕ = c1(L) ∈ Pic
(
PQg,n
)
⊗ Q. Denote by λ the pullback of the
Hodge class λ1 = detπ∗ωg,n on Mg,n. Furthermore, denote by δ0deg, δ
∞
deg, δirr, δj,k the classes
in Pic
(
PQg,n
)
⊗ Q of the corresponding divisors. We will also use the standard notation ψi,
i = 1, . . . , n, for the tautological classes on Mg,n as well as for their pullbacks to PQg,n.
Combining the results of [1, Theorem 2] and, e.g., [5, Theorem 3.3(b)], we get
Lemma 1.1. The rational Picard group Pic
(
PQg,n
)
⊗Q is freely generated over Q by the classes
ϕ, λ, ψi, δirr, δj,k, where i, k = 1, . . . , n, j = 0, . . . , [g/2], and 2 < 2j + k < 2g + n− 2.
To each pair (C, q) one can canonically associate a twofold branched cover f : Ĉ → C and
an abelian differential v on Ĉ, where Ĉ =
{
(x, v(x)) |x ∈ C, v(x) ∈ T ∗
xC, v(x)
2 = q(x)
}
. For
a generic (C, q) ∈ Qg,n the curve Ĉ is smooth of genus ĝ = 4g − 3 + n and v is holomorphic.
The covering f is invariant under the canonical involution (x, v(x)) 7→ (x,−v(x)) on Ĉ that
we denote by µ. Zeros and poles of q of odd order are branch points of the covering f . The
abelian differential v has second order zeros at simple zeros of q, and at the simple poles of q
the differential v is holomorphic and nonvanishing.
Consider the map p̂ : PQg,n → Mĝ, (C, q) 7→ Ĉ. This map induces a vector bundle p̂∗Λ1
ĝ →
Qg,n of dimension ĝ = 4g−3+n, where Λ1
ĝ → Mĝ is the Hodge vector bundle. The involution µ
on Ĉ induces an involution µ∗ on the vector bundle p̂∗Λ1
ĝ. Hence we have a decomposition
p̂∗Λ1
ĝ = Λ+ ⊕ Λ−,
where Λ+ (resp. Λ−) is the eigenbundle corresponding to the eigenvalue +1 (resp. −1) of µ∗.
Clearly, Λ+ = p∗π∗ωg is the pullback of the Hodge bundle on Mg, where p : PQg,n → Mg is
a natural projection (forgetful map). We call Λ− the Prym bundle. Its fibers are the spaces of
Prym differentials on Ĉ and have dimension 3g− 3+n. We call λP = c1(Λ−) ∈ Pic
(
PQg,n
)
⊗Q
the Prym class (slightly abusing the notation, we often denote the line bundles and their classes
in the Picard group by the same symbols).
In this paper we prove the following generalization of Theorem 1 in [11] to the case n > 0:
Tau Function and Moduli of Meromorphic Quadratic Differentials 3
Theorem 1.2. The Hodge class λ and the Prym class λP can be expressed in terms of the
tautological class ϕ and the classes δ0deg, δ
∞
deg and δDM by the formulas
λ =
(
5(g − 1)
36
− n
36
)
ϕ+
1
72
δ0deg −
1
18
δ∞deg +
1
12
δDM,
λP =
(
11(g − 1)
36
+
5n
36
)
ϕ+
13
72
δ0deg +
5
18
δ∞deg +
1
12
δDM.
On the other hand, according to [11, formula (5.16) and explanations thereafter],
λP = λ2 −
1
2
(3g − 3 + n)ϕ, (1.1)
where λ2 = detπ∗ω
2
g,n. This implies
Corollary 1.3. The following relation holds in Pic
(
PQg,n
)
⊗Q:
λ2 − 13λ = nϕ+ δ∞deg − δDM. (1.2)
Furthermore, combining (1.2) with Mumford’s formula
λ2 − 13λ =
n∑
i=1
ψi − δDM (1.3)
(cf., e.g., [2]), formula (7.8)), we get the following
Corollary 1.4. The class δ∞deg is expressed via the sum of ψ-classes and class ϕ as follows:
δ∞deg = −nϕ+
n∑
i=1
ψi . (1.4)
Remark 1.5. An analytic approach to the classes ψi was outlined, in particular, in [12].
The paper is organized as follows. In Section 2 we introduce a twofold canonical cover
corresponding to a pair consisting of a complex algebraic curve and a quadratic differential on
it with n simple poles. The main objective of this section is to discuss the action of the covering
involution on (co)homology of the cover and the associated matrix of b-periods. In Section 3 we
define two tau functions corresponding to the eigenvalues ±1 of the covering map, discuss their
basic properties and interpret them as holomorphic sections of line bundles on the moduli space
of quadratic differentials with simple poles. In Section 4 we study the asymptotic behavior of
the tau functions near the divisor D∞
deg and use it to express the Hodge and Prym classes via the
classes of the boundary divisors and the tautological class (the asymptotics of the tau functions
near the divisors D0
deg and Dj,k were thoroughly studied in [11]).
2 Geometry of the double cover
Let f : Ĉ → C be the double cover defined by the meromorphic quadratic differential q with
simple zeros {x1, . . . , x4g−4+n} and simple poles {y1, . . . , yn} on a smooth curve C, and let
µ : Ĉ → Ĉ be the corresponding involution. The covering map f is ramified over xj , yk and
we put x̂j = f−1(xj), ŷk = f−1(yk); the points x̂j , ŷk are exactly the fixed points of µ. By µ∗
(resp. µ∗) we denote the involution induced by µ in homology (resp. in cohomology) of Ĉ. The
space Λ1
Ĉ
of holomorphic abelian differentials on Ĉ splits into two eigenspaces Λ+ and Λ− of
4 D. Korotkin and P. Zograf
complex dimension g and 3g − 3 + n respectively that correspond to the eigenvalues ±1 of µ∗.
We have a similar decomposition in the real homology of Ĉ: H1
(
Ĉ,R
)
= H+ ⊕ H−, where
dimH+ = 2g, dimH− = 6g − 6 + 2n. Following [4], we pick 8g − 6 + 2n smooth 1-cycles on Ĉ{
aj , a
∗
j , ãk, bj , b
∗
j , b̃k
}
, j = 1, . . . , g, k = 1, . . . , 2g + n− 3, (2.1)
in such a way that
µ∗aj = a∗j , µ∗bj = b∗j , µ∗ãk + ãk = µ∗b̃k + b̃k = 0,
and the intersection matrix is(
0 I4g−3+n
−I4g−3+n 0
)
(here Ik denotes the k × k identity matrix).
The projections f∗aj = f∗(a
∗
j ) and f∗bj = f∗(b
∗
j ) on C form a canonical basis in H1(C), while
the projections f∗ãk and f∗b̃k are trivial in H1(C).
Remark 2.1. To avoid complicated notation, we will use the same symbols for the cycles (2.1),
their homology classes in H1
(
Ĉ
)
, and their pushforwards in H1(C). In particular, the images
of (2.1) give rise to a canonical basis in H1(C).
Denote by {uj , u∗j , ũk} the basis of normalized abelian differentials on Ĉ associated with (2.1),
so that the action of µ∗ on Λ1
Ĉ
is given by the matrix
M =
0 Ig 0
Ig 0 0
0 0 −I2g−3+n
.
The differentials u+j = uj + u∗j , j = 1, . . . , g, provide a basis in the space Λ+, whereas a basis
in Λ− is given by 3g − 3 + n Prym differentials u−l , where
u−l =
{
ul − u∗l , l = 1, . . . , g,
ũl−g, l = g + 1, . . . , 3g − 3 + n.
We also introduce the bases in the spaces H+ and H−. The classes
α+
j =
1
2
(aj + a∗j ), β+j = bj + b∗j , j = 1, . . . , g,
form a symplectic basis in H+, whereas the classes
α−
l =
1
2
(al − a∗l ), β−l = bl − b∗l , l = 1, . . . , g,
α−
l = ãl−g, β−l = b̃l−g, l = g + 1, . . . , 3g − 3 + n
form a symplectic basis in H−. The basis {a+j , α
−
l , β
+
j , β
−
l }, j = 1, . . . , g, l = 1, . . . , 3g − 3 + n,
is related to the canonical basis (2.1) by means of a (non-integer) symplectic matrix
S =
(
T 0
0
(
T t
)−1
)
Tau Function and Moduli of Meromorphic Quadratic Differentials 5
with
T =
Ig Ig 0
Ig −Ig 0
0 0 I2g−3+n
. (2.2)
The differentials u+j , u
−
l are normalized relative to the classes α+
j , α
−
l respectfully in the sense
that ∫
α+
i
u+j = δij ,
∫
α−
k
u−l = δkl,
where i, j = 1, . . . , g and k, l = 1, . . . , 3g − 3 + n. The corresponding matrices of β-periods Ω+
and Ω− are given by
(Ω+)ij =
∫
β+
i
u+j , i, j = 1, . . . , g,
(Ω−)kl =
∫
β−
k
u−l , k, l = 1, . . . , 3g − 3 + n.
The matrix Ω̂ of b-periods of {uj , ũk, u∗j} with respect to the homology basis (2.1) on Ĉ is related
to Ω+ and Ω− by the formula
Ω̂ = T−1
(
Ω+ 0
0 Ω−
)(
T t
)−1
.
We proceed with bidifferentials and projective connections on the double covers. Let B̂(x, y)
denote the canonical (Bergman) bidifferential on Ĉ×Ĉ associated with the homology basis (2.1);
B̂(x, y) is symmetric, has the second order pole on the diagonal x = y with biresidue 1, and all
a-periods of B̂(x, y) on Ĉ vanish. We put
B+(x, y) = B̂(x, y) + µ∗yB̂(x, y), B−(x, y) = B̂(x, y)− µ∗yB̂(x, y),
(the subscript y at µ∗ means that we take the pullback with respect to the involution on the
second factor in Ĉ×Ĉ). The bidifferential B+(x, y) is the pullback of the canonical bidifferential
B(x, y) on C×C (normalized relative to the classes f∗aj , where f : Ĉ → C is the covering map).
The bidifferential B−(x, y) is called Prym bidifferential in [11].
Properties of the differentials B± are summarized in [11, Lemma 2]. Near the diagonal x = y
in Ĉ × Ĉ we have
B±(x, y) =
dζ(x)dζ(y)
(ζ(x)− ζ(y))2
+
1
6
SB±(ζ(x)) + · · · ,
where ζ(x) is a local parameter at x ∈ Ĉ, and two projective connections SB+ and SB− that are
related by
SB±(x) = S
B̂
(x)± 6µ∗yB̂(x, y)
∣∣
y=x
. (2.3)
Here SB− is the Prym projective connection.
Now we describe the dependence of bidifferentials and projective connections on the choice of
homology basis. Let σ be a symplectic (i.e., preserving the intersection form) transformation of
H1
(
Ĉ,Z
)
. In the canonical basis (2.1), σ is represented by an Sp(8g − 6 + 2n,Z)-matrix
(
A B
C D
)
6 D. Korotkin and P. Zograf
with square blocks of size 4g− 3+ n. The canonical bidifferential B̂σ(x, y) on Ĉ × Ĉ associated
with the new basis is
B̂σ(x, y) = B̂(x, y)− 2π
√
−1uuu(x)t
(
CΩ̂ +D
)−1
Cuuu(y),
where uuu = {uj , ũk, u∗j}t is the vector of holomorphic abelian differentials on Ĉ normalized with
respect to {aj , ãk, a∗j}, j = 1, . . . , g, k = 1, . . . , 2g − 3 + n.
If σ commutes with the involution µ∗, then σ = diag(σ+, σ−), where σ+ (resp. σ−) is a sym-
plectic transformation of H+ (resp. H−) with half-integer coefficients (note that the intersection
form on H1
(
Ĉ,R
)
is invariant with respect to µ∗). In the bases {α+
j , β
+
j }, j = 1, . . . , g, and
{α−
l , β
−
l }, l = 1, . . . , 3g − 3 + n, these transformations can be written as
σ± :=
(
A± B±
C± D±
)
. (2.4)
Transformation properties of basic holomorphic differentials, Prym matrix, differentials B±
and projective connections SB± are described in [11, Lemma 3].
3 Tau functions
Here we define the necessary tau functions and study their basic properties. Only slight modi-
fications are needed comparing to the case n = 0 [11].
3.1 Definition of τ±
Consider two meromorphic quadratic differentials SB± − Sv on Ĉ, where SB± are the Bergman
projective connections (2.3) and Sv is given by
Sv =
(
v′
v
)′
− 1
2
(
v′
v
)2
.
where the prime means the derivative with respect to a local parameter ζ(x) (in other words,
Sv is the Schwarzian derivative of the abelian integral
∫ x
x0
v). Take the trivial line bundle on the
principal stratum Q
(
14g−4+n,−1n
)
and consider two connections
d± = d+ ξ±
with
ξ± = −
6g−6+2n∑
i=1
(∫
s∗i
φ±
)
d
∫
si
v,
where {si} = {αi, βi} and {s∗i } = {βi,−αi}, i = 1, . . . , 3g − 3 + n, being the dual bases in H−
as above, and the meromorphic abelian differentials φ± are given by
φ± = − 2
π
√
−1
SB± − Sv
v
.
The connections d± are flat, see [6, 8, 11] for details.
Definition 3.1. The tau functions τ± are (locally) covariant constant sections of the trivial line
bundle on the principal stratum Q
(
14g−4+n,−1n
)
with respect to the connections d±, that is,
d±τ± = 0. (3.1)
Explicit formulas for the tau functions τ± can be derived similar to [6, 11], but we are not
going to use them here.
Remark 3.2. The definition (3.1) follows the convention of [11]. The tau functions defined
by (3.1) are equal to the 48th power of the tau functions defined in [6, 8].
Tau Function and Moduli of Meromorphic Quadratic Differentials 7
3.2 Transformation properties of τ±
For our purposes we need to analyze transformation properties of τ±. The group C∗ of nonzero
complex numbers acts by multiplication on quadratic differential q on C and, therefore, on the
tau functions τ±. Under the action of ϵ ∈ C∗ the tau functions τ± transform like τ±(C, ϵq) =
ϵκ±τ±(C, q), where
κ+ =
20(g − 1)− 4n
3
, κ− =
44(g − 1) + 20n
3
. (3.2)
The formulas (3.2) follow from formulas (6.54) and (6.55) of [8] taking into account an extra
factor of 48 due to a different definition of τ± adopted here, see also (4.4).
Let us now describe the behavior of the tau functions under a change of the canonical homol-
ogy basis that commutes with the action of the involution µ. Note that τ+ (resp. τ−) is uniquely
determined by the subspace in H+ (resp. H−) generated by the classes α+
i (resp. α−
j ).
Let σ be a symplectic transformation of H1
(
Ĉ,R
)
commuting with µ∗ that is given by the
matrices σ± in the basis {α+
i , β
+
i , α
−
j , β
−
j }, i = 1, . . . , g, j = 1, . . . , 3g − 3 + n, as in (2.4). Then
the tau functions τ± transform under the action of σ by the formula
τσ±
τ±
= γ±(σ±) det(C±Ω± +D±)
48, (3.3)
where γ±(σ±)
3 = 1.
The proof of this statement for n > 0 is similar to the case n = 0 considered in [11, Theorem 3].
Namely, the transformation property (3.3) (with some constants γ±) follows from the differential
equations (3.1) and the transformation rule of the Bergman projective connections SB± under
the change of canonical bases in H±. The proof that γ3± = 1 uses explicit formulas for τ± which
use the distinguished local parameters near the ramification points of the cover C → Ĉ. The
details can be found in [8], see formulas (6.46) and (6.50) therein.
The following statement is an immediate consequence of (3.2) and (3.3):
Theorem 3.3. For the tau function τ+ (resp. τ−) its 3rd power τ3+
(
resp. τ3−
)
is a nowhere
vanishing holomorphic section of the line bundle λ144 ⊗ L20(g−1)−4n
(
resp. λ144P ⊗ L44(g−1)+20n
)
on the (projectivized) principal stratum PQ
(
14g−4+n,−1n
)
, where L denotes the tautological line
bundle on PQ
(
14g−4+n,−1n
)
.
Corollary 3.4. The Hodge and Prym classes in the rational Picard group of PQ
(
14g−4+n,−1n
)
satisfy the relations
λ =
5(g − 1)− n
36
ϕ, λP =
11(g − 1) + 5n
36
ϕ,
where ϕ = c1(L).
4 The Hodge and Prym classes on PQg,n
Here we compute the divisor classes of the tau functions τ± viewed as holomorphic sections of
line bundles on the compactification PQg,n. The boundary PQg,n \ PQ
(
14g−4+n,−1n
)
consists
of divisors D0
deg, D
∞
deg and DDM. The asymptotic behaviour of τ± near D0
deg and DDM for n > 0
is the same as for n = 0, the case considered in detail in [11].
Namely, in terms of transversal local coordinate t near DDM we have
τ± = t4(const + o(1)) as t→ 0, (4.1)
cf. formulas (5.10) and (5.12) in [11].
8 D. Korotkin and P. Zograf
In terms of a local transversal coordinate t near D0
deg
τ+ = t2/3(const + o(1)), τ− = t26/3(const + o(1)) (4.2)
as t→ 0, cf. Lemmas 8 and 9 in [11].
Therefore, it remains to analyze the asymptotics of τ± near the divisor D∞
deg.
4.1 Local behavior of tau functions near D∞
deg
Recall that the boundary component D∞
deg is the closure in PQg,n of the union of n copies of the
set Q
(
14g−5+n, 0,−1n−1
)
of equivalence classes of pairs (C, q), where C is a smooth curve with
a marked point and q is a quadratic differential with n− 1 labeled simple poles and 4g − 5 + n
simple zeros (the marked point is neither zero nor pole of q and distinguish the point where
a zero and a pole of q coalesce).
Consider a one parameter family (Ct, qt) of generic quadratic differentials transversal to D
∞
deg
that converges to (C0, q0) ∈ D∞
deg as t→ 0 (we can actually keep Ct fixed and ignore the rational
tail of C0). Without loss of generality we can assume that under such a degeneration a simple
zero xt1 and a simple pole yt1 of qt on Ct coalesce as t → 0. Denote by st ∈ H− the vanishing
cycle on Ĉ going around the path connecting the preimages x̂t1, ŷ
t
1 ∈ Ĉ, and denote by t =
∫
st
vt
the corresponding homological coordinate.
Assuming that the zero xt1 and the pole yt1 are close to each other, consider a small disk
Ut ⊂ Ct that contains x
t
1, y
t
1 and no other singularities of qt.
Lemma 4.1. The local coordinate t =
∫
st
vt is transversal to D∞
deg in a tubular neighborhood of
D∞
deg ⊂ Qg,n.
Proof. The proof is similar to that of Lemma 8 in [11]. Namely, one can choose a coordinate ζ
in Ut such that
q(ζ) =
ζ − ζ
(
xt1
)
ζ − ζ
(
yt1
)dζ2.
Then
t =
∫
st
vt = 2
∫ ζ(yt1)
ζ(xt
1)
(
ζ − ζ
(
xt1
)
ζ − ζ
(
yt1
))1/2
dζ = const
(
ζ
(
xt1
)
− ζ
(
yt1
))
.
Therefore, t =
∫
st
vt is transversal to D
∞
deg. ■
Lemma 4.2. The tau functions τ±(Ct, qt) have the following asymptotic behaviour as t→ 0:
τ+(Ct, qt) ∼ t−8/3τ+(C0, q0), τ−(Ct, qt) ∼ t40/3τ−(C0, q0). (4.3)
Proof. First, let us notice that for an arbitrary stratum Q(d1, . . . , dk), where d1, . . . , dk are inte-
gers not less than −1 and
∑k
i=1 di = 4g−4, the tau functions τ± have the following homogeneity
property: τ±(C, ϵq) = ϵκ±τ±(C, q) with
κ+ =
∑
i
di(di + 4)
di + 2
, κ− = κ+ + 6
∑
di odd
1
di + 2
, (4.4)
(see equations (6.54) and (6.55) of [8], as well as Remark 3.2). Now, in the limit t→ 0, we have
for some γ±
τ± ∼ tγ±τ±(C0, q0).
Tau Function and Moduli of Meromorphic Quadratic Differentials 9
To compute γ± we evaluate the differences of the homogeneity coefficients (4.4) on the strata
Q
(
14g−4+n,−1n
)
and Q
(
14g−5+n, 0,−1n
)
, i.e., when a pole and a zero annihilate each other (all
other contributions remain the same):
κ+(q)− κ+(q0) =
1 · 5
3
+
(−1) · 3
1
= −4
3
,
κ−(q)− κ−(q0) = κ+(q)− κ+(q0) + 6
(
1
3
+ 1
)
=
20
3
.
Since the homogeneity coefficient of the coordinate t is 1/2, we have γ+ = −8/3 and γ− =
40/3. ■
4.2 Proof of Theorem 1.2 and its consequences
Combining Theorem 3.3 with the asymptotics (4.1), (4.2) and (4.3) of τ± near DDM, D0
deg
and D∞
deg, we get the formulas
48λ− 20(g − 1)− 4n
3
ϕ =
2
3
δ0deg −
8
3
δ∞deg + 4δDM, (4.5)
48λP − 44(g − 1) + 20n
3
ϕ =
26
3
δ0deg +
40
3
δ∞deg + 4δDM, (4.6)
which immediately imply Theorem 1.2.
Excluding δ0deg from (4.5) and (4.6), we obtain
Corollary 4.3. The Prym class λP on PQg,n is expressed in terms of the Hodge class λ, the
tautological class ϕ and the boundary classes DDM, D∞
deg by the formula
λP − 13λ = δ∞deg − δDM − 1
2
(3g − 3− n)ϕ.
Furthermore, using the formula λP = λ2 − 1
2(3g − 3 + n)ϕ that relates the Prym class to the
class λ2 = π∗ω
2
g,n (see (1.1)), we get
Corollary 4.4. The following relation holds:
λ2 − 13λ = nϕ+ δ∞deg − δDM.
Combining this relation with the pullback to PQg,n of Mumford’s formula (1.3), we obtain
the equation (1.4)
δ∞deg = −nϕ+
n∑
i=1
ψi.
On the other hand, excluding δ∞deg from (4.5) and (4.6), we get the formula
δ0deg = 72λ+ 4
n∑
i=1
ψi − (10(g − 1) + 2n)ϕ− 6δDM
that coincides with the formula (1.2) in [11] for n = 0.
Acknowledgements
This work was supported by the Ministry of Science and Higher Education of the Russian
Federation, agreement no. 075-15-2019-1620. We thank A. Zorich for useful discussions. We
are grateful to anonymous referees for carefully reading the manuscript and pointing out several
typos and other inconsistencies.
10 D. Korotkin and P. Zograf
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1 Introduction and statement of results
2 Geometry of the double cover
3 Tau functions
3.1 Definition of tau_{pm}
3.2 Transformation properties of tau_{pm}
4 The Hodge and Prym classes on PQ_{g,n}
4.1 Local behavior of tau functions near D^{infty}_{deg}
4.2 Proof of Theorem 1.2 and its consequences
References
|
| id | nasplib_isofts_kiev_ua-123456789-211544 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T23:50:37Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Korotkin, Dmitry Zograf, Peter 2026-01-05T12:31:09Z 2022 Tau Function and Moduli of Meromorphic Quadratic Differentials. Dmitry Korotkin and Peter Zograf. SIGMA 18 (2022), 001, 10 pages 1815-0659 2020 Mathematics Subject Classification: 14H15; 14H70; 14K20; 30F30 arXiv:2108.01419 https://nasplib.isofts.kiev.ua/handle/123456789/211544 https://doi.org/10.3842/SIGMA.2022.001 The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most 𝑛 simple poles on genus 𝑔 complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials. This work was supported by the Ministry of Science and Higher Education of the Russian Federation, agreement no. 075-15-2019-1620. We thank A. Zorich for useful discussions. We are grateful to anonymous referees for carefully reading the manuscript and pointing out several typos and other inconsistencies. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Tau Function and Moduli of Meromorphic Quadratic Differentials Article published earlier |
| spellingShingle | Tau Function and Moduli of Meromorphic Quadratic Differentials Korotkin, Dmitry Zograf, Peter |
| title | Tau Function and Moduli of Meromorphic Quadratic Differentials |
| title_full | Tau Function and Moduli of Meromorphic Quadratic Differentials |
| title_fullStr | Tau Function and Moduli of Meromorphic Quadratic Differentials |
| title_full_unstemmed | Tau Function and Moduli of Meromorphic Quadratic Differentials |
| title_short | Tau Function and Moduli of Meromorphic Quadratic Differentials |
| title_sort | tau function and moduli of meromorphic quadratic differentials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211544 |
| work_keys_str_mv | AT korotkindmitry taufunctionandmoduliofmeromorphicquadraticdifferentials AT zografpeter taufunctionandmoduliofmeromorphicquadraticdifferentials |