Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve
Let be a smooth projective curve over a field of characteristic zero, and let be a non-empty set of rational points of . We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on ( , ) and the motivic classes of moduli stacks of semistable parabolic...
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| description | Let be a smooth projective curve over a field of characteristic zero, and let be a non-empty set of rational points of . We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on ( , ) and the motivic classes of moduli stacks of semistable parabolic Higgs bundles on ( , ). As a by-product, we give a criterion for the non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 070, 49 pages
Motivic Donaldson–Thomas Invariants of Parabolic
Higgs Bundles and Parabolic Connections on a Curve
Roman FEDOROV †, Alexander SOIBELMAN ‡ and Yan SOIBELMAN §
† University of Pittsburgh, Pittsburgh, PA, USA
E-mail: fedorov@pitt.edu
‡ Aarhus University, Aarhus, Denmark
E-mail: asoibel@qgm.au.dk
§ Kansas State University, Manhattan, KS, USA
E-mail: soibel@math.ksu.edu
Received November 19, 2019, in final form July 10, 2020; Published online July 27, 2020
https://doi.org/10.3842/SIGMA.2020.070
Abstract. Let X be a smooth projective curve over a field of characteristic zero and let D
be a non-empty set of rational points of X. We calculate the motivic classes of moduli
stacks of semistable parabolic bundles with connections on (X,D) and motivic classes of
moduli stacks of semistable parabolic Higgs bundles on (X,D). As a by-product we give
a criteria for non-emptiness of these moduli stacks, which can be viewed as a version of the
Deligne–Simpson problem.
Key words: parabolic Higgs bundles; parabolic bundles with connections; motivic classes;
Donaldson–Thomas invariants; Macdonald polynomials
2020 Mathematics Subject Classification: 14D23; 14N35; 14D20
1 Introduction and main results
1.1 Overview
Let k be a field of characteristic zero and X be a smooth geometrically connected projective
curve over k (geometric connectedness means that X remains connected after the base change to
an algebraic closure of k). In [12] we calculated the motivic classes of moduli stacks of semistable
Higgs bundles on X. These motivic classes are closely related to Donaldson–Thomas invariants,
see [22, 23]. In [12] we also calculated the motivic classes of moduli stacks of vector bundles with
connections on X by relating them to the motivic classes of stacks of semistable Higgs bundles.
In this paper, we extend these results to the parabolic case. Some of our results are parallel
to the results of A. Mellit in the case of finite fields (see [30]). One difference is that we fix the
eigenvalues of the residues. Another difference is that by working over a field of characteristic
zero, we are also able to treat bundles with connections. We also note that the calculation of the
motivic class requires subtler techniques, than the calculation of the volume of the corresponding
stack over a finite field.
1.2 Moduli stacks
Let us briefly describe the moduli stacks whose motivic classes we will be interested in. There
will be three classes of stacks.
This paper is a contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mu-
lase for his 65th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Mulase.html
mailto:fedorov@pitt.edu
mailto:asoibel@qgm.au.dk
mailto:soibel@math.ksu.edu
https://doi.org/10.3842/SIGMA.2020.070
https://www.emis.de/journals/SIGMA/Mulase.html
2 R. Fedorov, A. Soibelman and Y. Soibelman
1.2.1 Parabolic bundles with connections
Let D ⊂ X(k) be a non-empty set of rational points of X. A parabolic bundle of type (X,D) is
a collection E = (E,E•,•), where E is a vector bundle over X and Ex,• is a flag in its fiber Ex
for x ∈ D:
Ex = Ex,0 ⊇ Ex,1 ⊇ · · · ⊇ Ex,l ⊇ · · · , Ex,l = 0 for l� 0.
A connection on E with poles bounded by D is a morphism of sheaves of abelian groups ∇ : E →
E ⊗ ΩX(D) satisfying Leibniz rule. (Here ΩX is the canonical line bundle on X.) In this
case for x ∈ D one defines the residue of the connection resx∇ ∈ End(Ex). Let ζ = ζ•,• =
(ζx,j) be a sequence of elements of k indexed by D × Z>0 such that ζx,j = 0 for j � 0. Let
Conn(X,D, ζ) denote the moduli stack parameterizing collections (E,E•,•,∇), where (E,E•,•)
is a parabolic bundle of type (X,D), ∇ is a connection on E with poles bounded by D such
that (resx∇ − ζx,j1)(Ex,j−1) ⊂ Ex,j for all x ∈ D and j > 0. We usually skip X and D from
the notation as they are fixed, denoting Conn(X,D, ζ) simply by Conn(ζ). We call the points
of Conn(ζ) parabolic bundles with connections of type (X,D) with eigenvalues ζ.
For a parabolic bundle E = (E,E•,•) define the class of E as the following collection of
integers:
(rkE,dimEx,j−1 − dimEx,j ,degE) ∈ Z≥0 × Z≥0[D × Z>0]× Z. (1.1)
We also set rk E := rkE.
The stack Conn(ζ) decomposes according to the classes of parabolic bundles; denote the
component corresponding to parabolic bundles of class γ by Connγ(ζ). We will see that this
stack is an Artin stack of finite type over k. One of our main results (see Section 1.4 and
Theorem 8.8) is the calculation of the motivic class of this stack.
1.2.2 Parabolic Higgs bundles
Let ζ be as above. A parabolic Higgs bundle with eigenvalues ζ is a triple (E,E•,•,Φ), where
(E,E•,•) is a parabolic bundle of type (X,D), Φ: E → E⊗ΩX(D) is a morphism of OX -modules
(called a Higgs field on (E,E•,•)) such that for all x ∈ D and j > 0 we have
(Φ− ζx,j1)(Ex,j−1) ⊂ Ex,j ⊗ ΩX(D)x.
Denote the category and the stack of such Higgs bundles by Higgs(ζ). Unfortunately, this
stack is not of finite type over k, and in fact, has an infinite motivic volume. To resolve the
problem we endow the category with a stability structure. Let σ = σ•,• be a sequence of real
numbers indexed by D × Z>0. Let κ ∈ R≥0. We define the (κ, σ)-degree of a parabolic bundle
E = (E,E•,•) by
degκ,σ E := κ degE +
∑
x∈D
∑
j>0
σx,j(dimEx,j−1 − dimEx,j) ∈ R.
If E 6= 0, we define the (κ, σ)-slope of E as degκ,σ E/ rk E.
We say that a sequence σ = σ•,• of real numbers indexed by D×Z>0 is a sequence of parabolic
weights if for all x ∈ D we have
σx,1 ≤ σx,2 ≤ · · · (1.2)
and for all x and j we have σx,j ≤ σx,1 + 1. Let σ be a sequence of parabolic weights. Let
E = (E,E•,•) be a parabolic bundle. Let F ⊂ E be a saturated vector subbundle (that is,
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 3
E/F is torsion free). Set Fx,j := Fx ∩ Ex,j . Then F := (F, F•,•) is a parabolic bundle. We
say that a Higgs bundle (E,Φ) is σ-semistable, if for all saturated subbundles F of E preserved
by Φ the (1, σ)-slope of the corresponding parabolic bundle F is less than or equal to that of E.
We have an open substack Higgsσ−ss
γ (ζ) of Higgsγ(ζ) classifying σ-semistable parabolic Higgs
bundles. This stack is of finite type over k; we will calculate its motivic class (see Section 1.4
and Theorem 7.5).
We note that condition (1.2) is imposed on σ to ensure that we have Harder–Narasimhan
filtrations for parabolic Higgs bundles. We also note that, scaling κ and σ by the same positive
real number scales all the slopes by the same number. This is why we restrict to the case κ = 1
above (see Remark 3.10 for more details).
1.2.3 Semistable parabolic bundles with connections
We can also impose stability conditions on parabolic bundles with connections. Moreover, for
non-resonant connections we can work with more general stability conditions, than those for
Higgs bundles defined in the previous paragraph. A sequence ζ as above is called non-resonant
if for all x ∈ X and all i, j > 0 we have ζx,i − ζx,j /∈ Z6=0. Take κ ∈ R≥0 and a sequence σ of
real numbers indexed by D × Z>0 and satisfying condition (1.2).
Assume that ζ is non-resonant. We define (κ, σ)-semistability of parabolic bundles with
connections similarly to semistability of Higgs bundles but using the (κ, σ)-slope. Denote the
corresponding moduli stack by Conn(κ,σ)−ss
γ (ζ); this is an open substack of Connγ(ζ). If ζ
is resonant, then σ has to satisfy some additional conditions (see Proposition 8.12 and Re-
mark 8.14).
1.3 Motivic Donaldson–Thomas invariants
Our formulas for motivic classes of the moduli stacks above are all given in terms of certain
motivic classes Bγ called motivic Donaldson–Thomas invariants (see Section 1.5 for this termi-
nology), which we are going to define. First of all, we recall that in [12, Section 2] we defined
(following earlier works [11, Section 1], [19], and [22]) the ring of motivic classes of Artin stacks
denoted Mot(k). We also defined its dimensional completion Mot(k). For an Artin stack S of
finite type over k we have its motivic class [S] ∈ Mot(k). We denote its image in Mot(k) by the
same symbol.
For a curve X and a partition λ we defined the series Jmot
λ (z), Hmot
λ (z) ∈ Mot(k)[[z]] in [12,
Section 1.3.2]. The definitions (especially of Hmot
λ (z)) are somewhat long, so we will not recall
them here inviting the reader to look into [12]. We only note that Jmot
λ (z) and Hmot
λ (z) are
defined in terms of the motivic zeta-function of X (cf. (3.3) below). In particular, they only
depend on X but not on D. In this paper, we will denote them by Jmot
λ,X (z) and Hmot
λ,X (z)
respectively to emphasize that they depend on the curve X and to ensure that they are not
confused with motivic modified Macdonald polynomials H̃mot
λ (w•; z) and with motivic Hall–
Littlewood polynomials Hmot
λ (w•) defined below.
The modified Macdonald polynomials H̃λ(w•; q, z) are symmetric functions in variables w• =
(w1, w2, . . . ) with coefficients in Z[q, z]. In [30, Definition 2.5] the modified Macdonald poly-
nomials are defined as symmetric functions with coefficients in Q[q, z] but it is well-known
that the coefficients are integers (see, e.g., [14] and references therein). Note that, formally
speaking, symmetric functions are not polynomials (they become polynomials upon plugging in
wN+1 = wN+2 = · · · = 0). Let L =
[
A1
k
]
be the motivic class of the affine line. We denote by
H̃mot
λ (w•; z) the symmetric function with coefficients in Mot(k)[z] obtained from H̃λ(w•; q, z) by
substituting L for q. We denote their images in the ring of symmetric functions with coefficients
in Mot(k)[z] by H̃mot
λ (w•; z) as well.
4 R. Fedorov, A. Soibelman and Y. Soibelman
Let Γ+ denote the commutative monoid of sequences (r, r•,•, d), where r is a nonnegative
integer, r•,• is a sequence of nonnegative integers indexed by D × Z>0, d is an integer, subject
to the following conditions:
(i) For all x ∈ D we have
∞∑
j=1
rx,j = r. In particular, rx,j = 0 for j large enough.
(ii) If r = 0, then d = 0 (and so rx,j = 0 for all x and j).
The operation on Γ+ is the componentwise addition. For γ = (r, r•,•, d) ∈ Γ+ we set rk γ = r.
The significance of the monoid Γ+ is that the class of a parabolic bundle E defined by (1.1) is
an element of Γ+. We also need a submonoid Γ′+ ⊂ Γ+ given by d ≤ 0. Consider the completed
monoid ring Mot(k)[[Γ′+]], we write its elements as
∑
γ∈Γ′+
Aγeγ , where Aγ ∈ Mot(k), eγ are basis
vectors. It is convenient to identify eγ with a monomial
wr
∏
x∈D
∞∏
j=1
w
rx,j
x,j z
d,
where w•,• = (wx,j) is a sequence of variables indexed by D×Z>0. Then we identify Mot(k)[[Γ′+]]
with a subring of Mot(k)
[[
w,w•,•, z
−1
]]
. Similarly, we consider the completed monoid ring
Mot(k)[[Γ′+]]. We note that these rings are closely related to completed quantum tori considered
in [22, 23]. In our case, they are commutative, essentially because we are working with 2-
dimensional Calabi–Yau categories; see Sections 1.5 and 3.3 for more details.
Finally, we need the notion of plethystic exponent and logarithm. Let Mot(k)[[Γ′+]]0 denote
the subset of Mot(k)[[Γ′+]] consisting of elements with zero constant terms. Then we have
a bijection
Exp : Mot(k)[[Γ′+]]0 → 1 + Mot(k)[[Γ′+]]0
called the plethystic exponent. We refer the reader to Section 3.4 for the definition. Let the
plethystic logarithm Log be the inverse bijection. Let us write
L · Log
(∑
λ
w|λ|Jmot
λ,X
(
z−1
)
Hmot
λ,X
(
z−1
) ∏
x∈D
H̃mot
λ
(
wx,•; z
−1
))
=
∑
γ∈Γ′+
Bγeγ ,
where the sum in the LHS is over all partitions. We call the elements Bγ the Donaldson–Thomas
invariants. Note that B0 = 0.
When X = P1
k, we can define motivic Donaldson–Thomas invariants Bγ by a simpler formula
valid in Mot(k):
L · Log
∑
λ
w|λ|
∏
x∈D
H̃mot
λ
(
wx,•; z
−1
)
∏
h∈Hook(λ)
(
La(h) − z−l(h)−1
)(
La(h)+1 − z−l(h)
)
=
∑
γ∈Γ′+
Bγeγ , (1.3)
where Hook(λ) stands for the set of hooks of λ, a(h) and l(h) stand for the armlength and the
leglength of the hook h respectively. We show that for X = P1 the images of Bγ in Mot(k) are
equal to Bγ .
1.4 Explicit formulas
The following explicit formulas for the motivic classes are parts of Theorem 8.8, Theorem 7.5,
and Theorem 8.16 respectively. Let γ = (r, r•,•, d) ∈ Γ+, γ 6= 0. Let ζ be as in Section 1.2.1. For
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 5
κ ∈ k we define the (κ, ζ)-degree and the (κ, ζ)-slope of parabolic bundles similarly to (κ, σ)-
degree and (κ, σ)-slope defined in Section 1.2.3; the only difference is that the (κ, ζ)-degree and
(κ, ζ)-slope take values in k.
For each τ ∈ k, define the elements Cγ(ζ) ∈ Mot(k), where γ ranges over elements of Γ′+
such that γ = 0 or the (1, ζ)-slope of γ is τ , by the following formula
∑
γ∈Γ′+
deg1,ζ γ=τ rk γ
L−χ(γ)Cγ(ζ)eγ = Exp
∑
γ∈Γ′+
deg1,ζ γ=τ rk γ
Bγeγ
,
where χ(γ) := (g − 1)r2 +
∑
x∈D
∑
j<j′
rx,jrx,j′ , g is the genus of X. Let γ ∈ Γ+ be such that
deg1,ζ γ = 0. Then, according to Theorem 8.8, the stack Connγ(ζ) is of finite type over k and
we have in Mot(k)
[Connγ(ζ)] = C(r,r•,•,d−Nr)(ζ),
whenever N is large enough. If deg1,ζ γ 6= 0, then the stack Connγ(ζ) is empty.
Next, assume that ζ and σ are as in Section 1.2.2, for τ ∈ R define the elements Hγ(ζ, σ) ∈
Mot(k) by the following formula
∑
γ∈Γ′+
deg0,ζ γ=0
deg1,σ γ=τ rk γ
L−χ(γ)Hγ(ζ, σ)eγ = Exp
∑
γ∈Γ′+
deg0,ζ γ=0
deg1,σ γ=τ rk γ
Bγeγ
.
Assume that deg0,ζ γ=0, where γ ∈ Γ′+. Then, according to Theorem 7.5, the stackHiggsσ−ss
γ (ζ)
is of finite type over k and we have[
Higgsσ−ss
γ (ζ)
]
= H(r,r•,•,d−Nr)(ζ, σ),
whenever N is large enough. If deg0,ζ γ 6= 0, then the stack Higgsσ−ss
γ (ζ) is empty.
Finally, assume that ζ and (κ, σ) are as in Section 1.2.3. For τ ∈ k, τ ′ ∈ R define the elements
Cγ(ζ,κ, σ) ∈ Mot(k) by the following formula
∑
γ∈Γ′+
deg1,ζ γ=τ rk γ
degκ,σ γ=τ ′ rk γ
L−χ(γ)Cγ(ζ,κ, σ)eγ = Exp
∑
γ∈Γ′+
deg1,ζ γ=τ rk γ
degκ,σ γ=τ ′ rk γ
Bγeγ
.
Let γ ∈ Γ+ be such that deg1,ζ γ = 0. Then, according to Theorem 8.16, we have[
Conn(κ,σ)−ss
γ (ζ)
]
= C(r,r•,•,d−Nr)(ζ,κ, σ),
whenever N is large enough. If deg1,ζ γ 6= 0, then the stack Conn(κ,σ)−ss
γ (ζ) is empty.
If X = P1, we get similar results valid in Mot(k), by replacing Bγ with Bγ defined by a simpler
formula (1.3).
6 R. Fedorov, A. Soibelman and Y. Soibelman
Remark 1.1. We note that each of the above motivic classes depends only on finitely many
DT-invariants. Indeed, [Connγ(ζ)],
[
Higgsσ−ss
γ (ζ)
]
, and
[
Conn(κ,σ)−ss
γ (ζ)
]
depend only on Bγ′
with rk γ′ ≤ rk γ and for a given γ there are only finitely many such γ′ ∈ Γ′+.
Remark 1.2. We note also that all the stacks whose motivic classes we are calculating are of
finite type over k, so their motivic classes are defined in Mot(k). However, we can only calculate
their motivic classes in Mot(k) except when X = P1. The reason is that, our calculation is
based on the calculation of motivic classes of stacks of vector bundles on X (without parabolic
structures) with nilpotent endomorphisms. This calculation is performed in [12] and is, in turn,
based on the motivic analogue of Harder’s residue formula (see [12, Theorem 1.5.1 and Section 4]
and [15, Theorem 2.2.3]). This formula, which is essentially saying that “all vector bundles have
essentially the same motivic number of Borel reductions” involves some limiting process and is,
therefore, only valid in the completed ring Mot(k).
1.5 Aftermath
In Section 1.3 we defined the classes Bγ . These classes should be thought of as the Donaldson–
Thomas invariants of the stack Higgs(0) of parabolic Higgs bundles with nilpotent residues.
Note that this stack is the cotangent bundle of Bunpar(X,D), while the stacks Higgs(ζ) and
Conn(ζ) are twisted cotangent bundles. We emphasize that Bγ do not depend on ζ, κ, and σ. The
meaning of the formulas in Section 1.4 is that the Donaldson–Thomas invariants of these twisted
cotangent bundles are obtained by restricting the range of γ to the submonoid deg0,ζ γ = 0 in
the case of Higgs(ζ) and to the submonoid deg1,ζ γ = 0 in the case of Conn(ζ).
Another feature of the formulas is that the motivic classes of the stacks depend on equations
satisfied by κ and σ rather than on inequalities. In other words, there is no wall-crossing in
our case. This is not very surprising, as the category Higgs(0), being a cotangent bundle of
Bunpar(X,D), is a 2-dimensional Calabi–Yau category, cf. [36].
One can speculate that similar results should be valid for the twisted cotangent stacks to
the moduli stack of objects of any reasonable 1-dimensional category. Note that such cotangent
stacks were studied by G. Dobrovolska, V. Ginzburg, and R. Travkin in [9].
Another example of such a twisted cotangent stack is the category of vector bundles with
irregular connections and appropriate level structures. This example is certainly more compli-
cated as the corresponding abelian category has infinite homological dimension. We hope to
return to this question in subsequent publications.
The formulas in Section 1.4 are explicit but complicated. However, one can see that the
motivic classes under considerations belong to the sub-λ-ring of Mot(k) generated by L, X, and
the inverses of Li − 1 for i ≥ 1. We note that this ring is probably strictly larger, than the
subring of Mot(k) generated by L, the symmetric powers X(i), and the inverses of Li − 1 for
i ≥ 1. The reason is that Mot(k) is unlikely to be a special λ-ring, see Section 1.7. On the other
hand, if X = P1, then all our motivic classes are rational functions in L with denominators
being products of Li − 1 for i ≥ 1.
1.6 Other results
It is clear from the above formulas that we have a lot of equalities between different motivic
classes of Higgs bundles and bundles with connections. In particular, we show in Propositions 9.1
and 9.2 that every motivic class of the form
[
Higgsσ−ss
γ (ζ)
]
or
[
Conn(κ,σ)−ss
γ (ζ)
]
is equal to
some motivic class of the form [Connγ(ζ)], provided that k is not a finite extension of Q. As
a consequence, we derive from results of Crawley-Boevey [8] a criterion of non-emptiness of
our moduli stacks. It is not difficult to see that if X 6= P1
k, then the stack
[
Higgsσ−ss
γ (ζ)
]
is
non-empty if and only if deg0,ζ γ = 0, while the stack
[
Conn(κ,σ)−ss
γ (ζ)
]
is non-empty if and only
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 7
if deg1,ζ γ = 0. For X = P1
k the question is much more subtle and is related to the so-called
Deligne–Simpson problem. This problem was originally stated for k = C in [40]. It may be
reformulated for an arbitrary algebraically closed field k of characteristic 0 as follows: given
a sequence of glr-conjugacy classes C• indexed by D, does there exist a pair (E,∇) consisting of
a rank r vector bundle and a connection ∇ on E with poles bounded by D such that Resx∇ ∈ Cx
for all x ∈ D? Bundles with connections (E,∇) parameterized by Connγ
(
P1
k, D, ζ
)
are exactly
bundles with connections such that each residue Resx∇ lies in the closure of a conjugacy class
determined by γ and ζ (see [7]). If this conjugacy class is semisimple for each x ∈ D, then the
elements of Connγ
(
P1
k, D, ζ
)
are the solutions of the corresponding Deligne–Simpson problem.
For a comprehensive survey of the Deligne–Simpson problem, see [7, 24, 42].
We note that if k is not algebraically closed, one can ask a subtler question of whether there
is a k-rational point in
[
Higgsσ−ss
γ (ζ)
]
or
[
Conn(κ,σ)−ss
γ (ζ)
]
. We do not know the answer to this
question. A somewhat similar question for moduli spaces of quiver representations is considered
by V. Hoskins and F. Schaffhauser in [18].
At this point, we would like to emphasize that working with not necessarily algebraically
closed fields is inevitable in the motivic setup: even if one is only interested in the case k = C,
one still has to consider all fields of characteristic zero, see Remark 2.2 below.
One of the motivations for this work is the non-abelian Hodge theory of C. Simpson (see [39]).
In this paper, Simpson constructs an equivalence between a category of parabolic bundles with
connections and a category of Higgs bundles. In Proposition 9.4, we show that the corresponding
stacks have equal motivic classes. We note that neither statement can be derived from the other
(cf. [12, Remark 1.2.2]). We note also, that it is clear from our results that there are many
more equalities of motivic classes, than those that one can guess from the non-abelian Hodge
theory. We remark that V. Hoskins and S. Pepin Lehalleur have shown in [17, Theorem 4.2] that
the Voevodsky motives of the coarse moduli spaces of bundles with connections and of Higgs
bundles are equal in the case when the rank and the degree are coprime. One can ask whether
this can be upgraded to the parabolic situation.
1.7 Other relations with previous work
We have already noted that our results are closely related with the results of Mellit [30]. One
difference is that Mellit counts the weighted number of points over a finite field, while we
work over a field of characteristic zero and calculate motivic classes. Mellit counts the volumes
of moduli stacks of Higgs bundles but is not considering bundles with connections. Another
difference is that Mellit is not fixing the eigenvalues of Higgs fields.
On the other hand, Mellit’s answers are simpler as they do not involve Schiffmann’s polyno-
mials Hλ. In fact, Mellit’s simplification of Schiffmann’s formula from [37] has nothing to do
with parabolic structures. This simplification is the content of Mellit’s papers [29, 30, 31]. We
believe that this simplification does not go through in the motivic case because Mellit is using the
fact that the λ-ring structure on symmetric functions is special (which, roughly speaking, means
that λ•(xy) and λ•(λ•(x)) can be expressed in terms of λ•(x) and λ•(y)). It is known (see [26])
that the Grothendieck λ-ring of varieties is not a special λ-ring. We do not know whether the
Grothendieck λ-ring of stacks Mot(k) and its completion Mot(k) are special. In any case, if one
replaces Mot(k) by some its quotient that is a special λ-ring, then one expects that the Mellit’s
simplifications are valid in this quotient. Examples of such quotients are the Grothendieck ring
of the category of Chow motives and the maximal special quotient of Mot(k) (see [26]).
The paper [6], although conjectural, contains an alternative approach to the problem via
upgrading the computation of the motivic class of Higgs bundles to the problem about motivic
Pandharipande–Thomas invariants on the non-compact Calabi–Yau 3-fold associated with the
spectral curve.
8 R. Fedorov, A. Soibelman and Y. Soibelman
Note also that Mozgovoy and Schiffmann in [34] consider Higgs bundles with a twist by an
arbitrary line bundle of degree at least 2g− 2, where g is the genus of X. However, they do not
consider parabolic structures and do not fix eigenvalues.
Finally we note that the general philosophy of Donaldson–Thomas invariants and the ap-
proach via motivic and cohomological Hall algebras (see [22, 23]) are applicable to our situation.
For more details about the approach that uses motivic Hall algebras we refer the reader to [12,
Section 1.6, Remark 3.6.3].
1.8 Organization of the article
In Section 3 we define the category Bunpar(X,D) of parabolic bundles and its graded stack of
objects denoted by the same letter. Most of our stacks below will be stacks over Bunpar(X,D).
In Section 4 we study the stack of bundles with endomorphisms. This stack is the main
intermediate object in our calculations. First, we calculate the motivic classes of stacks of
parabolic bundles with nilpotent endomorphisms with fixed generic type. The calculation is
based on Theorem 4.4, saying that these motivic classes are products of motivic classes of
similar stacks without parabolic structures and of “local stacks” independent of the curve. This
is a motivic analogue of [30, Theorem 5.6]. However, the proof in the motivic case is significantly
more involved and, hopefully, more conceptual.
The motivic classes of stacks of vector bundles (without parabolic structures) with nilpotent
endomorphisms are calculated in the proof of [12, Theorem 1.4.1]. Since the motivic classes of
“local stacks” are independent of the curve, it is enough to calculate them for X = P1; we do
this using the ideas and results of Mellit [30].
Then we use the formalism of plethystic powers to calculate the motivic classes of stacks of
parabolic bundles with arbitrary endomorphisms.
In Section 5 we study parabolic Higgs bundles with fixed eigenvalues. If the eigenvalues are
equal to zero, then the endomorphisms of a given parabolic bundle and the Higgs fields on this
bundle are parameterized by vector spaces whose dimensions differ by some Euler characteristic.
Thus, it is easy to relate the motivic classes of the two stacks. If the eigenvalues are not zero,
then not every parabolic bundle admits a Higgs fields with these eigenvalues. We give a criterion
for existence of such a Higgs field in Lemma 5.3. This allows us to express the motivic class
of parabolic Higgs bundles with fixed eigenvalues in terms of the motivic class of the so-called
isoslopy parabolic bundles with endomorphisms (see Proposition 5.4).
The motivic class of isoslopy parabolic bundles with endomorphisms is derived from the
results of Section 4 with the help of a factorization formula (see Proposition 5.6). This is
analogous to [12, Proposition 3.5.1].
In Section 6 we use a version of Kontsevich–Soibelman factorization formula to calculate the
motivic classes of stacks of semistable Higgs bundles. These depend on two sets of parame-
ters: the eigenvalues and the stability condition. Somewhat surprisingly, these two sets come
symmetrically in the answer.
Up to Section 7 we work with nonpositive vector bundles, that is, vector bundles having no
subbundle of positive degree. Without stability this restriction is inevitable as otherwise the
moduli stacks would have infinite motivic volume. With a stability condition we can drop this
technical restriction; the motivic classes of semistable parabolic Higgs bundles whose underlying
vector bundles are not necessarily nonpositive are calculated in Section 7.
In Section 8 we study the moduli stack of bundles with connections – with or without sta-
bility condition. The strategy for connections is similar to that for Higgs bundles except that
the corresponding stacks are of finite type over k even without stability conditions. So we
first calculate the motivic classes of bundles with connections with given eigenvalues without
stability conditions and without nonpositivity assumptions and then use a version of Kontsevich–
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 9
Soibelman factorization formula to calculate the motivic classes of stacks of semistable bundles
with connections.
In Section 9 we give a precise criterion for non-emptiness of moduli stacks of Higgs bundles,
bundles with connections, or semistable bundles with connections. The idea is that such a stack
is non-empty if and only if its motivic class is non-zero. Using our explicit formulas we re-
write each of our motivic classes as the motivic class of a stack of bundles with connections
(without stability conditions). The non-emptiness of such a stack is decided using Lemma 8.2
and Crawley-Boevey’s result [8, p. 1334, Corollary].
2 Preliminaries
2.1 Conventions
We denote by k a field of characteristic zero. We denote by X a smooth projective geometrically
connected curve over k (recall that geometric connectedness means that X remains connected
after the base change to an algebraic closure of k). We denote by D a set of k-rational points
of X and by degD the number of elements of D.
If E is a vector space or a vector bundle, we denote by E∨ the dual vector space (resp. vector
bundle). We identify vector bundles with their sheaves of sections. If F is a coherent sheaf, we
denote by End(F ) the sheaf of its endomorphisms, we have End(F ) = F∨ ⊗ F if F is a vector
bundle.
2.1.1 Partitions and nilpotent matrices
By a partition we mean a non-increasing sequence of integers λ = λ1 ≥ λ2 ≥ · · · , where λl = 0
for l� 0. Set |λ| :=
∑
i λi. For partitions λ and µ we set 〈λ, µ〉 =
∑
i λ
′
iµ
′
i, where λ′ and µ′ are
the conjugate partitions.
We denote by glr,k or simply by glr the set of r × r matrices with entries in k. We say
that a nilpotent matrix n ∈ gl|λ| = gl|λ|,k is of type λ, if for all i ≥ 1 we have dim Kerni −
dim Kerni−1 = λi. For each partition λ choose a nilpotent matrix nλ of type λ. For concreteness,
we can take for nλ the direct sum of nilpotent Jordan blocks, where the number of blocks of size
i× i is equal to λi − λi+1.
A sequence (w1, . . . , wl, . . . ) we denote by w•.
2.1.2 Stacks
We will be working with stacks. All our stacks will have affine stabilizers. Our stacks will
be Artin stacks locally of finite type over a field except in Section 4.3, where we will have to
work with stacks whose points have stabilizers of infinite type. For a stack S we often abuse
notation by writing s ∈ S to mean that s is an object of the groupoid S(k), or an object of
the groupoid S(K), where K is an extension of k. Following [27, Chapter 5] we say that a K ′-
point ξ′ of S is equivalent to a K ′′-point ξ′′ of S if there is an extension K ⊃ k and k-embeddings
K ′ ↪→ K, K ′′ ↪→ K such that ξ′K is isomorphic to ξ′′K (as an object of S(K)). The corresponding
equivalence classes are called points of S; the set of points is denoted by |S|. See [12, Section 2]
for more details.
We write “morphism of stacks” to mean “1-morphism of stacks”. We write “of finite type”
to mean “of finite type over k”.
10 R. Fedorov, A. Soibelman and Y. Soibelman
2.2 Motivic functions and motivic classes
Recall that in [12, Section 2] we defined (following [11, Section 1], [19], and [22]) the ring of
motivic classes of Artin stacks denoted Mot(k).
More generally, for an Artin stack X locally of finite type over k, we defined the Mot(k)-
module of motivic functions on X denoted Mot(X ). For a morphism f : X → Y we have the pull-
back homomorphism f∗ : Mot(Y)→ Mot(X ). The pushforward homomorphism f! : Mot(X )→
Mot(Y) is defined when f is of finite type. We also defined the ring of completed motivic classes,
denoted Mot(k), and Mot(k)-modules of completed motivic functions Mot(X ) with a (probably
non-injective) morphism Mot(X ) → Mot(X ). We also defined the pullbacks and the pushfor-
wards of completed motivic functions.
We usually work with Mot(k) but our final results are formulated in Mot(k).
We defined the notion of a constructible subset of a stack. If X → Y is a morphism of finite
type, and S ⊂ X is a constructible subset, we defined the motivic function [S → Y] ∈ Mot(Y).
Recall [12, Proposition 2.6.1]:
Proposition 2.1. Assume that we are given A,B ∈ Mot(X ) are such that for all field extensions
K ⊃ k and for all k-morphisms ξ : SpecK → X we have ξ∗A = ξ∗B. Then A = B.
Remark 2.2. The previous proposition is one of the reasons we have to work with arbitrary
fields. Indeed, even if we start with k = C, to be able to apply the proposition we have to
consider all finitely generated extensions of C; see, for example, Section 4.3.5.
In Section 9 we will need the following proposition.
Proposition 2.3. An Artin stack of finite type over k is non-empty if and only if its motivic
class in Mot(k) is not equal to zero.
Proof. The ‘if’ direction is obvious. For the other direction assume for a contradiction that S is
a non-empty Artin stack of finite type over k such that [S] = 0 ∈ Mot(k). This means that for all
m ∈ Z we have [S] ∈ Fm Mot(k), where F • is the dimensional filtration on Mot(k). According
to [25, Propositions 3.5.6 and 3.5.9] every Artin stack of finite type with affine stabilizers has
a stratification by global quotients of the form T/GLn, where T is a scheme. Thus, replacing S
with a stratification and clearing the denominators, we may assume that S is a disjoint union of
integral affine schemes. Recall from [12, Section 2.5] that Mot(k) is the localization of the K-ring
of varieties Motvar(k) with respect to the multiplicative set generated by L and Li − 1, where
i > 0. Thus, multiplying S by a certain product of these elements, we may assume that the
class of S in Motvar(k) belongs to the subgroup Fm−1 Motvar(k) generated by the classes of the
varieties of dimension at most m − 1, where m = dimS. Compactifying each top-dimensional
connected component of S and taking the resolution of singularities, we may assume that S is
the disjoint union of smooth projective k-varieties.
Recall that the Hodge–Deligne polynomial of a smooth projective variety Y is
dimY∑
p,q=0
(−1)p+qhp,q(Y )upvq,
where hp,q = dimHq(Y,∧pΩY ). This extends uniquely to a homomorphism E : Motvar(k) →
Z[u, v]. Clearly, E([Y ]) has degree 2m, if Y is a smooth projective variety of dimension m. On
the other hand, E([Y ]) has degree at most 2m − 2, if Y is any variety of dimension at most
m − 1. We see that, on the one hand E(S) has degree 2m, on the other hand it has degree
2m− 2. We come to contradiction. �
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 11
2.3 Principal bundles and special groups
Let H be an algebraic group of finite type over k. Recall that a principal H-bundle over a k-
stack B is a stack E together with a schematic smooth surjective morphism of finite type E → B
and an action a : H ×k E → E such that H acts simply transitively on the fibers of E → B.
More precisely, the simple transitivity means that the morphism (a, p2) : H × E → E ×B E is
an isomorphism, where p2 : H × E → E is the projection. A principal bundle is trivial if there
is an isomorphism E ≈ H × B compatible with the action and the projection to B. If E is
a principal H-bundle over B and B′ → B is a morphism, then one gets an induced principal
H-bundle E ×B B′ over B′. The group H is called special if every principal H-bundle E over
a scheme B of finite type over k is locally trivial over B in the Zariski topology.
The following lemma is standard, see, e.g., [3, Section 2.3].
Lemma 2.4. Let H be a special group and E → B be a principal H-bundle, where B is an Artin
stack of finite type over k. Then in Mot(k) we have [E] = [H][B].
Proof. The case when B is a scheme is easily proved by Noetherian induction. If B is a stack,
then, using again [25, Propositions 3.5.6 and 3.5.9], we may assume that B is a global quotient:
B = S/GLn, where S is a scheme. Then we have the cartesian diagram
E′ −−−−→ Ey y
S −−−−→ B,
where E′ = E ×B S and E = E′/GLn. Applying [12, Corollary 2.2.2] with S = Spec k, we
get [E′] = [E][GLn] and S = [B][GLn]. Next, E′ is a principal H-bundle over the scheme S so
[E′] = [H][S]. Combining these equations we easily get the required statement. �
Recall that a k-group U is unipotent if it can be embedded into a group of strictly upper
triangular matrices. Every unipotent subgroup is obtained from the additive group of the 1-
dimensional vector space by iterated extensions.
Lemma 2.5. Let H be an algebraic group of finite type over k and let U be a unipotent subgroup.
Assume that H/U is special. Then H is special. In particular, every unipotent group is special.
Proof. Assume first that H is a unipotent group. We claim that every principal H-bundle
over an affine scheme is trivial. We prove this by induction on dimH. If dimH = 1, then H is
the additive group and the principal H-bundles over a scheme B are classified by the coherent
cohomology group H1(B,OB), which vanishes as soon as B is affine. If dimH > 1, then there
is a subgroup H ′ ⊂ H such that dimH ′ < dimH and dim(H/H ′) < dimH. The groups H ′
and H/H ′ are unipotent. Recall that the principal H-bundles over B are classified by the
first non-abelian étale cohomology group H1
ét(B,H). Now the statement follows from the exact
sequence H1
ét(B,H
′) → H1
ét(B,H) → H1
ét(B,H/H
′). In particular, every unipotent group is
special.
Now assume that H/U is special, where U is unipotent, and consider, for an affine B, the
exact sequence 1 = H1
ét(B,U)→ H1
ét(B,H)→ H1
ét(B,H/U). We see that a principal H-bundle
is trivial over an affine scheme if the induced principal H/U -bundle is trivial. The lemma
follows. �
Lemma 2.6. Let Zλ be the centralizer of nλ in GL|λ|. Then Zλ is a special group.
Proof. It is well-known that the quotient of Zλ by its unipotent radical is the product of GLri
for some ri ∈ Z>0. It remains to note that GLri are special groups and the product of special
groups is special. �
12 R. Fedorov, A. Soibelman and Y. Soibelman
3 Parabolic bundles
3.1 Definitions and notations
Recall that k denotes a field of characteristic zero, X stands for a smooth projective geometrically
connected curve over k, and D is a set of rational points of X. We often assume that D 6= ∅; in
this case X has a divisor of degree one defined over k. We will often have to consider sequences
indexed by D × Z>0 or by D × Z≥0. A typical notation will be r•,•. If x ∈ D, then rx,• stands
for the sequence rx,1, rx,2, . . . (or rx,0, rx,1, . . . ).
The monoid of all sequences r•,• indexed by D × Z>0 with terms rx,j in a commutative
monoid S and such that rx,j = 0 for j � 0 (that is, functions on D × Z>0 with finite support)
will be denoted by S[D × Z>0].
Definition 3.1. A parabolic bundle of type (X,D) is a collection (E,E•,•), where E is a vector
bundle over X and Ex,• is a flag in Ex for x ∈ D:
Ex = Ex,0 ⊇ Ex,1 ⊇ · · · ⊇ Ex,l ⊇ · · · , Ex,l = 0 for l� 0.
We have the category Bunpar(X,D) of parabolic bundles. We sometimes denote a parabolic
bundle by a single boldface letter: E = (E,E•,•). The morphism from E to E′ is a morphism
ϕ : E → E′ such that for all x ∈ D and j ≥ 0 we have ϕ(Ex,j) ⊂ E′x,j . This category is an
additive k-linear category. The direct sum of (E,E•,•) and (E′, E′•,•) is (E ⊕ E′, E•,• ⊕ E′•,•).
We note that the decomposition of a parabolic bundle into a direct sum of indecomposable
parabolic bundles is unique up to isomorphism, while the isotypic summands are unique; the
proof is similar to [1, Theorem 3] (see also [12, Proposition 3.1.2]).
We often skip X and D from the notation, writing Bunpar instead of Bunpar(X,D).
Abusing notation, we denote by Bunpar the stack of objects of the category Bunpar. Precisely,
if S is a k-scheme, then Bunpar(S) is the groupoid of collections (E,E•,•), where E is a vector
bundle over S×kX, Ex,• is a filtration by vector subbundles of the restriction of E to S×k x for
x ∈ D. Here by a subbundle of E|S×kx we mean a subsheaf that splits off as a direct summand
Zariski locally over S.
3.2 Monoids Γ+ and Γ′
+
Consider the free abelian group Z× Z[D × Z>0]× Z and its subgroup Γ consisting of (r, r•,•, d)
such that for all x ∈ D we have
∑∞
j=1 rx,j = r.
Let Γ+ ⊂ Γ be the monoid of sequences (r, r•,•, d) such that
(i) r ≥ 0 and for all x ∈ D and j > 0 we have rx,j ≥ 0;
(ii) if r = 0, then d = 0.
Note that it follows from these conditions that r = 0 implies that (r, r•,•, d) is the zero
sequence; we denote it by 0. Define the class function:
cl : Bunpar → Γ+, (E,E•,•) 7→ (rkE,dimEx,j−1 − dimEx,j ,degE).
For γ = (r, r•,•, d) ∈ Γ we set rk γ := r. For a parabolic bundle E we set rk E := rk cl(E).
For γ ∈ Γ+, we denote by Bunpar
γ the stack of objects of class γ; this is an open and closed
substack of Bunpar. It is often convenient to think of Bunpar =
⊔
γ∈Γ+
Bunpar
γ as a Γ+-graded
stack. Note that Bunpar
0 has a single object: the parabolic bundle of rank zero.
Let γ = (r, r•,•, d) ∈ Γ+. The projection Bunpar
γ (X,D) → Bunr,d(X) to the stack of
rank r degree d vector bundles on X is schematic and of finite type (in fact, projective). Thus
Bunpar
γ (X,D) and Bunpar(X,D) are Artin stacks locally of finite type.
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 13
Let us call a vector bundle on X nonpositive if it does not have a subbundle of positive
degree. Recall that in [12, Section 3.2]) we called a vector bundle on X HN-nonnegative, if
its Harder–Narasimhan spectrum is nonnegative. by [12, Lemma 3.2.1(i)] there is an open
substack Bun+(X) ⊂ Bun(X) classifying HN-nonnegative vector bundles. Moreover, by [12,
Lemma 3.2.1(iii)] the substack of Bun+(X) corresponding to vector bundles of rank r and
degree d is of finite type (this substack was denoted by Bun≥0
r,d(X) in loc. cit.).
By [12, Lemma 3.2.1(ii)] a bundle E is HN-nonnegative if and only if it has no quotient
bundles of negative degree. Thus a vector bundle E is nonpositive if and only if its dual E∨ is
HN-nonnegative. The following lemma is now clear.
Lemma 3.2. There is an open substack Bun−(X) of Bun(X) classifying nonpositive vector
bundles. The assignment E 7→ E∨ is an isomorphism between Bun−(X) and Bun+(X). The
component of Bun−(X) corresponding to vector bundles of fixed degree and rank is of finite type.
Set
Bunpar,− = Bunpar,−(X,D) := Bunpar(X,D)×Bun(X) Bun−(X).
In other words, Bunpar,− is the stack (and the category) of parabolic bundles on X whose
underlying vector bundle is nonpositive. Set also
Bunpar,−
γ = Bunpar,−
γ (X,D) := Bunpar
γ (X,D)×Bun(X) Bun−(X).
Lemma 3.3. For all γ ∈ Γ+ the stack Bunpar,−
γ (X,D) is an Artin stack of finite type.
Proof. Follows from Lemma 3.2. �
Let Γ′+ be the submonoid of Γ+ consisting of sequences with d ≤ 0. Clearly, Bunpar,−
γ 6= ∅
only if γ ∈ Γ′+.
3.3 Categories over Bunpar and Γ+-graded stacks
We will consider below many categories with a forgetful functor to Bunpar (e.g., the category of
parabolic Higgs bundles). Let C be such a category and denote by C its stack of objects as well.
Assume that the morphism C → Bunpar is of finite type. Define the stacks
Cγ := C ×Bunpar Bunpar
γ , C− := C ×Bunpar Bunpar,−, C−γ := C ×Bunpar Bunpar,−
γ . (3.1)
By Lemma 3.3, the stack C−γ is of finite type. The stack C =
⊔
γ∈Γ+
Cγ is Γ+-graded, while the
stack C− =
⊔
γ∈Γ+
C−γ is Γ′+-graded. Moreover, the stack C− is of finite type as a graded stack,
that is, its graded components are stacks of finite type.
The group ring Mot(k)[Γ+] is closely related to quantum tori (cf. [22]). This has a natural
basis eγ , where γ ranges over Γ+; the multiplication is given by eγeγ′ = eγ+γ′ . The reason
for the multiplication in the quantum torus to be commutative is that we are actually working
with 2-dimensional Calabi–Yau categories, and hence the skewsymmetrization of the Euler form
vanishes (cf. [36]).
Let Mot(k)[[Γ+]] be the completion of Mot(k)[Γ+] (this can be viewed as the group of Mot(k)-
valued functions on Γ+).
If D is a Γ+-graded stack of finite type, we consider the generating series
[D] :=
∑
γ∈Γ+
[Dγ ]eγ ∈ Mot(k)[[Γ+]]. (3.2)
14 R. Fedorov, A. Soibelman and Y. Soibelman
We call [D] the graded motivic class of the stack D. Recall that in [22] the motivic Donaldson–
Thomas series was defined as an element of the completed motivic quantum torus. In our case,
it associates to a Γ+-graded stack D an infinite series in the commutative motivic quantum
torus corresponding to the monoid Γ+ endowed with the trivial bilinear form. Thus, it coincides
with [D].
Sometimes it is convenient to write eγ explicitly as
eγ = wr
∏
x∈D
∞∏
j=1
w
rx,j
x,j z
d ∈ Z
[[
w,w•,•, z, z
−1
]]
⊂ Mot(k)
[[
w,w•,•, z, z
−1
]]
,
where γ = (r, r•,•, d). Here w = w•,• stands for the collection of variables
wx,• := (wx,1, wx,2, . . . , wx,j , . . . ), x ∈ D.
The variables w, z, and wx,j for x ∈ D, j ≥ 1 are commuting variables.
Remark 3.4. Note that we do not fix the lengths of flags. Let us fix a function l : D → Z>0
and consider only flags of length at most l(x) at x. Let Γ+,l be the submonoid of Γ+ consisting
of sequences (r, r•,•, d) such that rx,j = 0 whenever j > l(x). We have the obvious projection
Γ+ → Γ+,l, which induces a homomorphism Πl : Mot(k)[[Γ+]]→ Mot(k)[[Γ+,l]]. Explicitly, this
is just setting wx,j = 0 whenever j > l(x). Let D be as in (3.2), then Πl[D] =
∑
γ∈Γ+,l
[Dγ ]eγ ∈
Mot(k)[[Γ+,l]] is the graded motivic class of the substack of D where the lengths of the flags are
bounded by l. We see that the difference between fixing the length of flags and allowing flags of
arbitrary lengths corresponds on the quantum torus side to the difference between polynomials
in infinite number of variables and polynomials in finite number of variables.
In our applications, [D] will be symmetric in each sequence of variables wx,• (cf. Remark 4.2).
In this case, the difference between fixing and not fixing the lengths corresponds on the side
of motivic classes to the difference between symmetric polynomials and symmetric functions,
cf. [28, Chapter 1, Section 2].
We emphasize that Mot(k)[[Γ+]] is not a ring. However, Mot(k)[[Γ′+]] ⊂ Mot(k)[[w,w•,•, z
−1]]
is a ring and [C−] ∈ Mot(k)[[Γ′+]] whenever C is a stack of finite type over Bunpar. This is in
accordance to the general theory in [22], where one fixes a strict sector in R2 in order to have
well-defined Donaldson–Thomas invariants. We can also replace Mot(k) with Mot(k) in all the
above constructions.
3.4 Motivic zeta-functions and plethystic operations
Following [21], for a variety Y define its motivic zeta-funcion by
ζY (z) :=
∞∑
n=0
[
Y (n)
]
zn ∈ Mot(k)[[z]], (3.3)
where Y (n) = Y n/Σn is the n-th symmetric power of Y (Σn denotes the group of permutations).
Consider the group (1+zMot(k)[[z]])×, where the group operation is multiplication. According
to [10, Theorem 2.3] ζ can be uniquely extended to a homomorphism
ζ : Mot(k)→ (1 + zMot(k)[[z]])× : A 7→ ζA(z)
such that we have ζLnA(z) = ζA
(
Lnz
)
for all n ∈ Z and A ∈ Mot(k). Clearly,
ζA(z) ≡ 1 +Ax (mod z2).
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 15
Thus we have equipped Mot(k) with a λ-ring structure. Note that Mot(k) is not a special
λ-ring, in particular, ζ(AB) cannot be expressed in terms of ζ(A) and ζ(B) (so some authors
would call this a pre-λ-ring structure).
According to loc. cit., this homomorphism ζ is continuous with respect to the dimensional
filtration on Mot(k), so it extends to a homomorphism
ζ : Mot(k)→
(
1 + zMot(k)[[z]]
)×
,
which coincides with the one constructed in [12, Section 1.3.1].
Let Mot(k)[[Γ′+]]0 stand for the series without constant term. We define the plethystic expo-
nent Exp: Mot(k)[[Γ′+]]0 → (1 + Mot(k)[[Γ′+]]0)× by
Exp
∑
γ∈Γ′+
Aγeγ
=
∏
γ∈Γ′+
Exp(Aγeγ) =
∏
γ∈Γ′+
ζAγ (eγ).
One shows easily that this is an isomorphism of abelian groups. Denote the inverse isomorphism
by Log (the plethystic logarithm). Finally, we define the plethystic power by
Pow:
(
1 + Mot(k)[[Γ′+]]0
)
×Mot(k)→ 1 + Mot(k)[[Γ′+]]0 : (f,A) 7→ Exp(ALog(f)).
We note that we can similarly define Exp, Log, and Pow for the completed ring Mot(k), which
coincide with the operations defined in [12] when D = ∅.
3.5 Parabolic subbundles and quotient bundles
Let E = (E,E•,•) be a parabolic bundle of type (X,D). We say that E′ = (E′, E′•,•) is a strict
parabolic subbundle of E if E′ is a saturated subbundle of E (that is, E/E′ is torsion free)
and for all x and j we have E′x,j = Ex,j ∩ E′x. Note that strict parabolic subbundles of E =
(E,E•,•) are in bijective correspondence with saturated subbundles of E. Let E′ = (E′, E′•,•)
be a strict parabolic subbundle of E = (E,E•,•); set E′′ := E/E′. Then we have a parabolic
structure on E′′ given by E′′x,j := Ex,j/E
′
x,j . We call the parabolic bundle E/E′ := (E′′, E′′•,•) the
quotient parabolic bundle of E. Thus, the quotient parabolic bundles of E are also in bijective
correspondence with saturated subbundles of E. Finally, in the above situation, we say that
0→ E′ → E→ E/E′ → 0 (3.4)
is a short exact sequence. We also say that E is an extension of E/E′ by E′. It is clear that in
this case we have cl(E) = cl(E′) + cl(E/E′). One can use the short exact sequences above to
define the group K0(Bunpar); the class function cl extends to cl : K0(Bunpar)→ Γ.
Remark 3.5. The category Bunpar is not abelian. It can be extended to an abelian category
by viewing vector bundles with flags as coherent sheaves on orbifold curves. Then the abelian
category is the category of coherent sheaves on this orbifold. This extension will not be used in
the current paper. On the other hand, if we define short exact sequences in Bunpar as sequences
isomorphic to some sequence of the form (3.4), then Bunpar becomes an exact category in the
sense of Quillen.
Let ϕ : E → F be a morphism of vector bundles on X. We say that ϕ is generically an isomor-
phism if it is an isomorphism at the generic point of X. Equivalently, ϕ is an isomorphism over
a non-empty Zariski open subset of X. Another reformulation is that ϕ is injective and F/ϕ(E)
is a torsion sheaf. Sometimes one says in this situation that E is a lower modification of F .
16 R. Fedorov, A. Soibelman and Y. Soibelman
Definition 3.6. We say that a morphism of parabolic bundles ϕ : (E,E•,•) → (F, F•,•) is
generically an isomorphism if the underlying morphism E → F is generically an isomorphism.
Lemma 3.7. Let ϕ : E→ F be a morphism of parabolic bundles. Then there are strict parabolic
subbundles E′ ⊂ E and F′ ⊂ F such that ϕ can be decomposed as
E
ϕ1−→ E/E′
ϕ2−→ F′
ϕ3−→ F,
where ϕ1 is the canonical projection, ϕ2 is generically an isomorphism, ϕ3 is the canonical
embedding.
Proof. Write E = (E,E•,•), F = (F, F•,•), let ϕ′ : E → F be the underlying morphism of
vector bundles. Note that Kerϕ′ is a vector subbundle of E. Indeed, E/Kerϕ′ is isomorphic
to a subsheaf of F , so it is torsion free. Let E′ be the strict subbundle of E whose underlying
vector bundle is Ker(ϕ′). Let F ′ be the saturation of the image of ϕ′ (that is, F ′ is the unique
saturated vector subbundle of F containing ϕ′(E) such that the quotient F ′/ϕ′(E) is a torsion
sheaf). Let F′ be the strict subbundle of F whose underlying vector bundle is F ′. Now the
existence of the decomposition is clear. �
3.6 Generalized degrees and slopes
Let A be a Q-vector space (in applications it will be k or R). Let κ ∈ A, ζ = ζ•,• ∈ A[D×Z>0].
Then we define the homomorphism degκ,σ : Γ→ A by
degκ,ζ(r, r•,•, d) = κd+
∑
x∈D
∑
j>0
ζx,jrx,j .
If rk γ 6= 0, we define the (κ, ζ)-slope of γ by degκ,ζ γ/ rk γ. We write degκ,ζ E for degκ,ζ cl(E)
and call degκ,ζ E/ rk E the (κ, σ)-slope of E.
We say that a parabolic bundle E is (κ, ζ)-isoslopy if the (κ, ζ)-slope of any direct summand
of E is equal to the (κ, ζ)-slope of E.
We remark that it is common to write ζ ? γ for deg0,ζ γ and degζ γ for deg1,ζ γ but we
prefer a uniform notation. We also remark that for an exact sequence (3.4) we have degκ,ζ E =
degκ,ζ E′ + degκ,ζ(E/E
′).
3.7 Parabolic weights and stability conditions
The following definition should be compared to [30, Definition 6.9].
Definition 3.8. We say that a sequence σ = σ•,• of real numbers indexed by D × Z>0 is
a sequence of parabolic weights if for all x ∈ D we have
σx,1 ≤ σx,2 ≤ · · · (3.5)
and for all x and j we have σx,j ≤ σx,1 + 1.
To every sequence of parabolic weights we will associate a notion of stability on parabolic
bundles in Definition 3.9 below. Thus we denote the set of all sequences of parabolic weights by
Stab = Stab(X,D).
Fix σ ∈ Stab.
Definition 3.9. A parabolic bundle E is σ-semistable if for all strict parabolic subbundles
E′ ⊂ E we have
deg1,σ E′
rk E′
≤
deg1,σ E
rk E
. (3.6)
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 17
Remark 3.10. We can similarly define semistability for any κ > 0 replacing the condition
σx,j ≤ σx,1 + 1 with σx,j ≤ σx,1 +κ. However, scaling κ and σ by the same positive real number
scales all the slopes by the same number, so we would get the same notion of semistability. This
is why we restrict to the case κ = 1 above. The case κ = 0 would not yield stacks of finite type;
however, the possibility of taking κ = 0 will be useful below, when we work with connections
(see Section 8.3).
Proposition 3.11. Let E ∈ Bunpar be a parabolic bundle. Then there is a unique filtration
0 = E0 ⊂ E1 ⊂ · · · ⊂ Em = E by strict parabolic subbundles such that all the quotients Ei/Ei−1
are σ-semistable and we have τ1 > · · · > τm, where τi is the (1, σ)-slope of Ei/Ei−1.
Proof. We start with a Lemma.
Lemma 3.12. Let the morphism E → F be generically an isomorphism. Then deg1,σ E ≤
deg1,σ F.
Proof. Write E = (E,E•,•) and F = (F, F•,•). Let ϕ : E → F be the underlying morphism of
vector bundles. For x ∈ D let dx denote the dimension of the kernel of ϕx. Then
degE = degF − length(F/ϕ(E)) ≤ degF −
∑
x∈D
dx.
On the other hand, for all x ∈ D and i > 0 we have dimEx,i ≤ dimFx,i + dx. Hence
deg1,σ E = degE +
∑
x,j>0
σx,j(dimEx,j−1 − dimEx,j)
= degE +
∑
x∈D
(
σx,1 rkE +
∑
i>0
(σx,i+1 − σx,i) dimEx,i
)
≤ degF −
∑
x∈D
dx +
∑
x∈D
(
σx,1 rkF +
∑
i>0
(σx,i+1 − σx,i)(dimFx,i + dx)
)
= deg1,σ F +
∑
x∈D
dx
(
−1 +
∑
i>0
(σx,i+1 − σx,i)
)
≤ deg1,σ F.
Lemma 3.12 is proved. �
The rest of the proof of Proposition 3.11 is completely analogous to the proof of [16, Sec-
tion 1.3] in view of Lemma 3.7. �
In the situation of Proposition 3.11(i) we say that the filtration is the Harder–Narasimhan
filtration of E (or HN-filtration for short) and τ1 > · · · > τm is the σ-HN spectrum of E.
We define Bunpar,≤τ and Bunpar,≥τ as full subcategories of Bunpar whose objects are parabolic
bundles with the σ-HN spectrum contained in (−∞, τ ] (resp. [τ,∞)). We emphasize that the
categories Bunpar,≤0 and Bunpar,− should not be confused with each other: they coincide only
if σ = 0. The following lemma is standard.
Lemma 3.13. Let E be an object of Bunpar,≤τ and E′ be an object of Bunpar,≥τ ′, where τ < τ ′.
Then HomBunpar(E′,E) = 0.
18 R. Fedorov, A. Soibelman and Y. Soibelman
4 Parabolic pairs
4.1 Parabolic pairs and their generic Jordan types
The notion of parabolic pair, interesting by itself, will be used as a technical tool for studying
parabolic Higgs bundles in Section 5 and parabolic bundles with connections in Section 8. Our
main results in this section are Theorem 4.19 and Corollary 4.21. They give explicit answers
for graded motivic classes of stacks of nilpotent parabolic pairs and parabolic pairs respectively.
We will also give a simplified answer in the case X = P1 in Section 4.7.
Definition 4.1. A parabolic pair (E,Ψ) consists of a parabolic bundle
E = (E,E•,•) ∈ Bunpar(X,D)
and an endomorphism Ψ of E (that is, an endomorphism of E preserving each Ex,j). If Ψ is
nilpotent we will speak about nilpotent parabolic pairs.
Parabolic pairs as well as nilpotent parabolic pairs form an additive k-linear category denoted
Pair = Pair(X,D) (resp. Pairnilp = Pairnilp(X,D)). Again, we abuse notation by denoting
the stacks of objects by the same symbols. We define Pairγ , Pairnilp
γ , Pair−γ , Pairnilp,−
γ etc.
following the general construction (3.1) of Section 3.3.
The forgetful morphisms Pair → Bunpar and Pairnilp → Bunpar are schematic and of finite
type. In particular, Pair− and Pairnilp,− are Γ′+-graded Artin stacks of finite type (in the
graded sense).
Let K ⊃ k be an extension and (E,E•,•,Ψ) ∈ Pairnilp(K). If we trivialize E at the generic
point of XK = X ×k SpecK, Ψ becomes a rkE × rkE nilpotent matrix. Its Jordan type is
a partition λ of rkE. Thus we get a locally closed stratification of Pairnilp according to the
generic Jordan type of the nilpotent endomorphism
Pairnilp(X,D) =
⊔
λ
Pairnilp(X,D, λ),
where the disjoint union is over all partitions. In other words, Pairnilp(X,D, λ) classifies nilpo-
tent parabolic pairs such that the endomorphism is generically conjugate to nλ (that is, conjugate
to nλ at the generic point of X, or, equivalently, at each point of a non-empty Zariski open subset
of X). We remark that any endomorphism generically conjugate to nλ is necessarily nilpotent.
Again, we define the Γ′+-graded stacks Pairnilp,−(X,D, λ) using the general formalism (3.1)
of Section 3.3.
4.2 Motivic classes of parabolic bundles with nilpotent endomorphisms
Our goal in this section is to calculate the graded motivic class (that is, the motivic Donaldson–
Thomas series, cf. [22])[
Pairnilp,−(X,D, λ)
]
=
∑
γ∈Γ′+
[
Pairnilp,−
γ (X,D, λ)
]
eγ
= w|λ|
∑
γ=(r,r•,•,d)∈Γ′+
[
Pairnilp,−
γ (X,D, λ)
]∏
x,j
w
rx,j
x,j z
d. (4.1)
The partition λ is fixed until the end of Section 4.3. This graded motivic class is calculated as
follows. First, in Section 4.3 we write this graded motivic class as the product of a term that is
independent of the parabolic structures, and the “local” terms independent of the curve. The
first term has been calculated in [12]. Since the local terms are independent of the curve, it is
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 19
enough to calculate them when X = P1. More precisely, we will work with P1 and two points
with parabolic structures (that is, D = {0,∞}) but we will calculate the sum over all partitions
(Section 4.4). This part is very similar to [30, Section 5.4]. In Section 4.6 we give the explicit
answer for the graded motivic classes under consideration. Using the formalism of plethystic
powers we then easily calculate the class of parabolic bundles with not necessarily nilpotent
endomorphisms.
Remark 4.2. We will see in Theorem 4.19 that (4.1) is a symmetric function in wx,• for each
x ∈ D. This can be explained as follows. Note that the “Weyl group” W :=
∏
x∈D
Σ∞ acts on Γ+
and Γ′+ in the obvious way (here Σ∞ is the inductive limit of the permutation groups Σl). Using
the commutativity of the motivic Hall algebra of the category of representations of the Jordan
quiver (the quiver with one vertex and one loop), one can easily show that the motivic classes[
Pairnilp,−
γ (X,D, λ)
]
are W -invariant. Thus, we can re-write (4.1) as
w|λ|
∑′
γ=(r,r•,•,d)∈Γ′+
[
Pairnilp,−
γ (X,D, λ)
] ∏
x∈D
mrx,•(wx,•)z
d,
where the summation is only over r•,• such that we have rx,j ≥ rx,j+1 for all x and j (that is,
rx,• is a partition of |λ|). Here, for a partition µ, mµ is the symmetric function equal to the sum
of all monomials whose ordered list of exponents is µ.
4.3 Factorization of graded motivic classes of stacks
of nilpotent parabolic pairs
In this section we factorize (4.1) as the product of the global part (depending only on X but
not on D) and the local parts corresponding to points of D (but independent of X). This is
a motivic version of [30, Theorem 5.6]. We follow the same ideas, though some parts of Mellit’s
proof do not work in the motivic case and must be replaced by different arguments. On the
other hand, we were able to simplify some parts of Mellit’s proof, in particular, by working with
stacks.
Note that Pairnilp,−(X,∅, λ) classifies pairs (E,Ψ), where E is a nonpositive vector bundle
on X, Ψ is an endomorphism of E generically conjugate to nλ but there are no parabolic
structures.
Lemma 4.3. The motivic class
[
Pairnilp,−
γ
(
P1,∅, λ
)]
∈ w|λ|Mot(k)
[[
z−1
]]
is invertible.
Proof. It is enough to show that the degree 0 part
[
Pairnilp,−
|λ|,0
(
P1,∅, λ
)]
is invertible in Mot(k).
Note that a nonpositive vector bundle of degree 0 on P1 is necessarily trivial. Thus, this degree
zero part classifies pairs (E,Ψ), where E is a trivial vector bundle and Ψ is a constant endo-
morphism conjugate to nλ. It is easy to see that this stack is isomorphic to the classifying stack
of the centralizer Zλ of nλ. By Lemma 2.6 Zλ is special. Now it follows from Lemma 2.4 or [12,
Lemma 2.2.3] that
[
Pairnilp,−
|λ|,0
(
P1,∅, λ
)]
= 1/[Zλ]. �
We need some notation. Note that
[
Pairnilp,−(P1,∞, λ
)]
∈ w|λ|Mot(k)
[[
w∞,•, z
−1
]]
. For
x ∈ D let[
Pairnilp,−(P1,∞, λ
)]
x
∈ w|λ|Mot(k)
[[
wx,•, z
−1
]]
denote the result of replacing w∞,• by wx,• in this series.
Theorem 4.4. We have in Mot(k)[[Γ′+]].
[
Pairnilp,−(X,D, λ)
]
=
[
Pairnilp,−(X,∅, λ)
] ∏
x∈D
[
Pairnilp,−(P1,∞, λ
)]
x[
Pairnilp,−(P1,∅, λ
)] .
20 R. Fedorov, A. Soibelman and Y. Soibelman
The proof of the theorem occupies the rest of Section 4.3; it is based on the local study of
stacks in the formal neighborhood of D.
The main idea of the proof is very simple. Let us assume that D = {x} is a single rational
point of X. In Section 4.3.2 we will define the stack Pairloc,fl classifying triples (F,Φ, F•),
where F is a rank |λ| vector bundle over the formal completion of X at x, Φ is a nilpotent
endomorphism of F generically conjugate to nλ, F• is a flag in Fx preserved by Φ(x). We have
an obvious restriction morphism Pairnilp,−(X,x, λ) → Pairloc,fl. We will see in Lemma 4.12
that this restriction morphism has constant fiber. Thus, one is tempted to write the graded
motivic class
[
Pairnilp,−(X,x, λ)
]
as the product of the graded motivic class of this fiber and
of Pairloc,fl. This would quickly lead to the proof of the theorem. Unfortunately, Pairloc,fl is
not an Artin stack as its points have inertia groups of infinite type, so its motivic class does not
make sense. The major part of the proof consists of going around this problem.
Let us give the overview of the proof. In Section 4.3.1 we define and study the schemes of
jets into gl|λ|. In Section 4.3.2 we study the local stacks; they are essentially the quotients of
the schemes of jets by the group of jets of GL|λ|. In Section 4.3.3 we re-write the theorem as
a statement about motivic classes of graded components. In Section 4.3.4 we study the fibers of
the localization map; this is the main part of the proof. We complete the proof in Section 4.3.5.
4.3.1 Jets
We will denote the non-archimedean local field k((t)) by K and its ring of integers k[[t]] by O.
The order of pole at t = 0 gives rise to a valuation map val : K→ Z∪{−∞}, where val(0) = −∞.
Clearly, val extends to glr,K as the maximum of valuations of all matrix elements. Let J(X)
denote the jet scheme of a scheme X (this is a scheme of infinite type), and let JN (X) denote
the scheme of order N − 1 jets. In particular, J1(X) = X.
For an algebraic group G of finite type over k we have the jet group GO := J(G) and the jet
group of finite type JN (G). The N -th congruence subgroup G(N) is the kernel of the projection
GO → JN (G). We also have the ind-group of loops GK containing GO. Let ∆ := SpecO be
the formal disc and ∆̊ := SpecK be its generic point (the punctured formal disc). Also set
∆N := Spec k[[t]]/tN . The groups GLr,O, GLr,K, and JN (GLr) are the groups of automorphisms
of the trivial vector bundles on ∆, ∆̊, and ∆N respectively. For more details on the jet and loop
groups we refer the reader to [43].
Set r = |λ|. Consider the orbit stratification of glr under the adjoint action of GLr: glr =⊔
µ`r
Oµ, where Oµ is the adjoint orbit containing nµ. Let Oµ denote the Zariski closure of Oµ.
Set
J(λ) := J
(
Oλ
)
−
⋃
Oµ⊂Oλ−Oλ
J
(
Oµ
)
.
Note that J(λ) parameterizes morphisms ∆→ glr such that the image of the generic point of ∆
is in Oλ, that is, jets that are generically conjugate to nλ.
Definition 4.5. We say that a loop g ∈ GLr,K(k) = GLr(K) is kernel-strict if g−1nλg ∈ glr,O
and g−1 induces an isomorphism between the O-modules Kernλ ⊗k O and Ker
(
g−1nλg
)
.
Remark 4.6. Note that our definition is a little different from [30, Definition 3.8]. Mellit’s
definition of kernel-strictness depends also on a choice of a matrix θ. In terminology of Mellit
our g is kernel-strict for θ = g−1nλg. Note also that the results of Mellit we are using here and
below are formulated over finite fields but are valid over any field, proofs being the same.
Let Φ be a k-point of J(λ), then there is a kernel-strict g ∈ GLr,K(k) = GLr(K) such that
Φ = g−1nλg. Set deg Φ := val(det g). The existence of such g and independence of the degree
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 21
on the choice of g is proved in [30, Lemma 3.7]. It follows also from loc. cit. that the degree is
nonnegative.
If K ⊃ k is a field extension, we similarly define the degree of a K-point of J(λ). The degree
is compatible with field extensions. Thus we get a stratification of the set of points of J(λ):
|J(λ)| =
⊔
d≥0
Jd(λ). Let πN : J(λ)→ JN
(
Oλ
)
be the truncation map. Set Jd,N (λ) := πN (Jd(λ)).
Proposition 4.7. For a nonnegative integer d and a partition λ there is a positive integer
N(d, λ) such that for N > N(d, λ) we have
(i) For all Φ in Jd(λ) there is a kernel-strict g with val(g) < N/2, val
(
g−1
)
< N/2 such that
gΦg−1 = nλ.
(ii) Jd(λ) = π−1
N (Jd,N (λ)).
(iii) Any two points in the same fiber of the projection Jd(λ) → Jd,N (λ) are conjugate by an
element of GLr,O.
Proof. By [30, Lemma 3.7] there is N0 ≥ 1 (depending on λ and d) such that for all Φ ∈ Jd(λ)
there is a kernel-strict g with val(g) < N0 such that gΦg−1 = nλ. Then val(det g) = d. Set
N1 := rN0 + d. Then by Cramer’s rule val
(
g−1
)
< N1. We also have val(g) < N1. Take
N(d, λ) := 4N1. With this choice of N(d, λ) part (i) of the proposition is clear.
Note that (ii) is saying that the degree of an infinite jet depends only on its N -th truncation
if N > 4N1. Thus to prove (ii) we need to show that if Φ ∈ Jd(λ) and Φ′ ∈ J(λ) is such that
Φ′ ≡ Φ (mod z4N1), then Φ′ ∈ Jd(λ). Choose a kernel-strict g with val(g) < N1, val
(
g−1
)
< N1
such that gΦg−1 = nλ. Then gΦ′g−1 ≡ nλ (mod z2N1). We need a lemma.
Lemma 4.8. There is g′ ∈ GL
(2N1)
r such that g′gΦ′g−1(g′)−1 = nλ.
Proof. Set Φ′′ := gΦ′g−1 and N2 := 2N1. Write Φ′′ ≡ nλ + ΦN2z
N2 (mod zN2+1). Since Φ′′ ∈
J
(
Oλ
)
, ΦN2 belongs to the tangent space to Oλ at nλ, which is naturally identified with [nλ, glr].
Thus there is gN2
∈ glr such that [nλ, gN2
] = ΦN2 . Then
(
1 + gN2
zN2
)
Φ′′
(
1 + gN2
zN2
)−1 ≡ nλ
(mod zN2+1).
Repeating this process we find gN2+1 , gN2+2 , . . . ∈ glr such that for j > 0 we have(
1 + gN2+jz
N2+j
)
· · ·
(
1 + gN2
zN2
)
Φ′′
(
1 + gN2
zN2
)−1 · · ·
(
1 + gN2+jz
N2+j
)−1
≡ nλ (mod zN2+j+1).
It remains to take g′ =
∞∏
j=0
(
1 + gN2+jz
N2+j
)
. Lemma 4.8 is proved. �
We return to the proof Proposition 4.7. Note that g−1g′g ∈ GLr,O. It is easy to see that the
set of kernel-strict loops is invariant under the multiplication by points of GLr,O on the right, so
g′g = g
(
g−1g′g
)
is kernel-strict. Clearly, val(det(g′g)) = val(det g) = d, so (ii) follows. Further,
g−1g′g conjugates Φ′ to Φ, so (iii) follows as well. Proposition 4.7 is proved. �
For every d ≥ 0 we fix N(d, λ) satisfying the conditions of the above proposition.
Definition 4.9. For N > N(d, λ) we call any jet in Jd,N (λ) stabilized and call d its degree.
According to the proposition, the degree of a stabilized jet is well-defined and any two lifts
of a stabilized jet to an infinite jet are conjugate by an element of GLr,O. Note that for every
jet Φ ∈ Jd(λ) its truncation πN (Φ) is stabilized for N large enough.
22 R. Fedorov, A. Soibelman and Y. Soibelman
4.3.2 Local stacks
Consider the quotient stack Pairloc = Pairloc(λ) := J(λ)/GLr,O, where GLr,O acts by conjuga-
tion. We skip λ from the notation as it is fixed until the end of Section 4.3. Since the degree
function on J(λ) is GLr,O-invariant, we get the degree function on the points of Pairloc.
Note that Pairloc classifies pairs (F,Φ), where F is a rank r = |λ| vector bundle over ∆,
Φ is a nilpotent endomorphism of F generically conjugate to nλ. This follows from the fact that
every vector bundle on ∆ is trivial. It also follows that every K-point of Pairloc is isomorphic
to a point of the form (Or,Φ), where Φ ∈ glr,O.
We emphasize that Pairloc is not an Artin stack (its isotropy groups are not of finite type).
Define Pairloc
N := JN
(
Oλ
)
/JN (GLr); this is an Artin stack of finite type. The points
of Pairloc
N are the pairs (F,Φ) where F is a vector bundle on ∆N , Φ is an endomorphism
of F such that if we trivialize F , Φ becomes a jet with values in Oλ. We say that (F,Φ) is
stabilized if Φ is stabilized in the sense of Definition 4.9. Note that this does not depend on
the trivialization of F . If (F,Φ) is stabilized, then we have a well-defined notion of the degree
of (F,Φ). Explicitly, we can lift (F,Φ) to a point (Or,Φ) of Pairloc, and the degree of (F,Φ) is
equal to the degree of Φ ∈ glr,O.
We define the stack Pairloc,fl as the stack classifying triples (F,Φ, F•), where (F,Φ) is a point
of Pairloc, F• is a flag in the fiber F0 preserved by Φ(0). We define the stack Pairloc,fl
N as the
stack classifying triples (F,Φ, F•), where (F,Φ) is a point of Pairloc
N , F• is a flag in F0 preserved
by Φ(0).
4.3.3 Preparation for the proof of Theorem 4.4
We will assume that D = x is a single rational point of X. This will unburden the notation; the
general case is proved similarly. Thus we want to prove that[
Pairnilp,−(X,x, λ)
][
Pairnilp,−(P1,∅, λ
)]
=
[
Pairnilp,−(X,∅, λ)
][
Pairnilp,−(P1,∞, λ
)]
x
.
Equating the graded components, we see that this reduces to the following proposition.
Proposition 4.10. Let d be a nonpositive integer, r• = (r1, r2, . . . ) be a sequence of nonnegative
integers such that
∑
i ri = r = |λ|. Then we have in Mot(k):∑
d′+d′′=d
[
Pairnilp,−
r,r•,d′
(X,x, λ)× Pairnilp,−
r,d′′
(
P1,∅, λ
)]
=
∑
d′+d′′=d
[
Pairnilp,−
r,d′ (X,∅, λ)× Pairnilp,−
r,r•,d′′
(
P1,∞, λ
)]
.
This proposition will be proved in Section 4.3.5. We emphasize that the sum is over all
d′, d′′ ∈ Z with d′ + d′′ = d but the terms are non-zero only if d′, d′′ ∈ [d, 0]. We note that the
RHS is manifestly independent of x. Thus, the LHS is independent of x as well.
4.3.4 The restriction to the formal neighborhood of x
We keep the simplifying assumption that D = {x} is a single point; we write r• instead of rx,•.
Fix γ = (r, r•, d) ∈ Γ′+. For x ∈ X let OX,x be the local ring of x and ÔX,x be its formal
completion. Set ∆x := Spec ÔX,x. Choose a formal coordinate at x, use it to identify ∆x
with ∆ and the N -th infinitesimal neighborhood ∆x,N of x with ∆N . Consider the restriction
morphism
locfl
x : Pairnilp,−
r,r•,d
(X,x, λ)→ Pairloc,fl
r• , (E,Ψ, Ex,•) 7→ (E|∆x ,Ψ|∆x , Ex,•). (4.2)
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 23
Similarly we have a morphism
locx : Pairnilp,−
r,d (X,∅, λ)→ Pairloc, (E,Ψ) 7→ (E|∆x ,Ψ|∆x). (4.3)
Our nearest goal is to describe the fibers of these morphisms. For a nonpositive integer e, let
Fibe(X,x) denote the open substack of Pairnilp,−
r,e (X,∅, λ) consisting of (E,Ψ) such that Ψ is
conjugate to nλ at x. Let Fibe(X,x) denote the stack of triples (E,Ψ, s), where (E,Ψ) is a point
of Fibe(X,x), s is a trivialization of E over ∆x such that Ψ = nλ in this trivialization. Recall
that in Section 4.3.2 we defined the notion of degree for the points of Pairloc.
Lemma 4.11.
(i) The fiber of locfl
x over (F,Φ, F•) is isomorphic to Fibd+e(X,x), where e is the degree
of (F,Φ).
(ii) Similarly, the fiber of locx over (F,Φ) is isomorphic to Fibd+e(X,x), where e is the degree
of (F,Φ).
Proof. We prove (ii) first. Fix a trivialization of F on the formal disc ∆. Then Φ becomes an
element of glr,K and we choose a kernel-strict g such that gΦg−1 = nλ. Then val(det g) = e.
Denote the fiber under consideration by Fib. The fiber can be described as the stack of triples
(E,Ψ, s), where E is a nonpositive vector bundle, Ψ is an endomorphism, s is the trivialization
of E over ∆x such that in this trivialization we have Ψ|∆x = Φ. Note that such Ψ is automatically
conjugate to nλ at the generic point of X.
If (E,Ψ, s) is a point of Fib, then E|X−x is trivialized over the punctured disc ∆̊x, and we
use the g chosen above to glue E|X−x with the trivial bundle kr ×∆x on ∆̊x (we recall that g
can be viewed as an automorphism of the trivial vector bundle on ∆̊x). We obtain a new vector
bundle E′ on X with an isomorphism E′|X−x ' E|X−x and a trivialization over ∆x. Thus Ψ
gives rise to an endomorphism Ψ′ of E′|X−x. It is easy to derive from the definition of g that in
the given trivialization we have Ψ′|∆̊x
= nλ. Thus Ψ′ extends to x and, moreover, in the given
trivialization of E′ over ∆x we have Ψ′|∆x = nλ.
Note that E′ is nonpositive. Indeed, Ker Ψ is nonpositive as a subbundle of E. Since g is
kernel-strict, the isomorphism between Ker Ψ and Ker Ψ′ extends from X−x to X. Thus Ker Ψ′
is also nonpositive. But by [30, Proposition 5.3] this implies that E′ is nonpositive as well.
Next, we have an isomorphism between ∧rE and ∧rE′ over X − x, and it has a zero of order
val(det g) at x. Thus degE′ = deg∧rE′ = deg∧rE + val(det g) = degE + e = d+ e.
We have constructed a morphism Fib→ Fibd+e(X,x). Conversely, given a point (E′,Ψ′, s′)
of Fibd+e(X,x), we use g and s′ to construct a new bundle E with an isomorphism to E′ over
X−x and a trivialization over ∆x. Then Ψ′|X−x give rise to an endomorphism of E|X−x and we
check that it extends to x and, moreover, in the trivialization of E over ∆x we have Ψ|∆x = Φ.
Now it is easy to see that the two constructions are inverse to each other. This proves (ii).
Now (i) follows from the cartesian diagram
Pairnilp,−
r,r•,d
(X,x, λ)
locfl
x−−−−→ Pairloc,fl
r•y y
Pairnilp,−
r,d (X,∅, λ)
locx−−−−→ Pairloc,
where the vertical arrows correspond to forgetting the flags. �
Consider now the compositions of (4.2) and (4.3) with restrictions to the N -th infinitesimal
neighborhood of x.
locfl
x,N : Pairnilp,−
r,r•,d
(X,x, λ)→ Pairloc,fl
r•,N
, (E,Ψ, Ex,•) 7→ (E|∆x,N
,Ψ|∆x,N
, Ex,•). (4.4)
24 R. Fedorov, A. Soibelman and Y. Soibelman
Similarly we have a morphism
locx,N : Pairnilp,−
r,d (X,∅, λ)→ Pairloc
N , (E,Ψ) 7→ (E|∆x,N
,Ψ|∆x,N
). (4.5)
Let (F,Φ, F•) be a point of Pairloc,fl
r•,N
and assume that (F,Φ) is stabilized. By Definition 4.9 we
can find Φ′ ∈ glr,O such that (Or,Φ′) lifts (F,Φ) and N > N(e, λ), where e is the degree of Φ′ and
N(e, λ) is the integer number from Proposition 4.7, which was fixed just before Definition 4.9.
Choose a kernel-strict g ∈ glr,K such that gΦ′g−1 = nλ. Denote by Z
(N)
g the intersection
ZO ∩
(
g−1GL(N)g
)
⊂ GLr,K, where Z = Zλ is the centralizer of nλ in GL|λ| (cf. Lemma 2.6).
Recall that by Proposition 4.7(i) we may assume that the orders of the poles of g and g−1 are less
than N/2 so we have Z
(N)
g ⊂ Z(1). This is a pro-unipotent group. Clearly, ZO (and thus Z
(N)
g
as well) acts on Fibd+e(X,x) by changing the trivialization of E on ∆x.
Lemma 4.12.
(i) Let (F,Φ, F•) ∈ Pairloc,fl
r•,N
be such that (F,Φ) is stabilized, choose g ∈ GLr,K as in the previ-
ous paragraph. Then the fiber of locfl
x,N over (F,Φ, F•) is isomorphic to Fibd+e(X,x)/Z
(N)
g ,
where e is the degree of (F,Φ).
(ii) Similarly, the fiber of locx,N over (F,Φ) is isomorphic to Fibd+e(X,x)/Z
(N)
g .
Proof. Let us prove (ii), the proof of (i) is completely analogous. Denote the fiber under
consideration by Fib and let Fib be the fiber of locx over (Or,Φ′). Then we have a restriction
morphism Fib → Fib. It follows from the stability of (F,Φ) and Proposition 4.7(iii) that this
morphism is surjective. On the other hand, it is easy to see that two points of Fib map to the
same point of Fib if and only if they differ by the action of an element of gZOg
−1 ∩ GL(N).
On the other hand, according to Lemma 4.11, Fib ' Fibd+e(X,x). One checks that under
this isomorphism, the action of gZOg
−1 ∩ GL(N) on Fib corresponds to the action of Z
(N)
g on
Fibd+e(X,x). �
We need to calculate the motivic class of this fiber. Set Zg := ZO/Z
(N)
g ; this is a group of
finite type.
Lemma 4.13. We have in Mot(k)[
Fibd+e(X,x)/Z(N)
g
]
= [Fibd+e(X,x)]/[Zg].
Proof. For large M we have Z(M) ⊂ Z(N)
g and this subgroup is normal.
Claim. The groups Z
(N)
g /Z(M), ZO/Z
(M), and ZO/Z
(N)
g are special.
Proof of the claim. Recall that Z
(N)
g ⊂ Z(1). The group Z
(N)
g /Z(M) is special, since every
unipotent group is special by Lemma 2.5. Next, the quotient of ZO/Z
(M) by the unipotent
subgroup Z(1)/Z(M) is equal to Z and the statement follows from Lemmas 2.5 and 2.6. A similar
argument shows that ZO/Z
(N)
g is special. �
We continue with the proof of Lemma 4.13. Next, Fibd+e(X,x)/Z(M) is a ZO/Z
(M)-principal
bundle over Fibd+e(X,x). Since ZO/Z
(M) is a special group, we get by Lemma 2.4[
Fibd+e(X,x)/Z(M)
]
=
[
ZO/Z
(M)
]
[Fibd+e(X,x)].
Similarly, Fibd+e(X,x)/Z(M) is a Z
(N)
g /Z(M)-principal bundle over Fibd+e(X,x)/Z
(N)
g and we
get [
Fibd+e(X,x)/Z(M)
]
=
[
Z(N)
g /Z(M)
][
Fibd+e(X,x)/Z(N)
g
]
.
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 25
Finally, ZO/Z
(M) is a Z
(N)
g /Z(M)-principal bundle over Zg and we have[
ZO/Z
(M)
]
=
[
Z(N)
g /Z(M)
]
[Zg].
The lemma follows from these three equations. �
Lemma 4.14. Let N be an integer larger than N(j, λ) for all j = 0, . . . ,−d. Assume that the
fiber of (4.5) over (F,Φ) is non-empty. Then (F,Φ) is stabilized. A similar statement holds for
the fibers of (4.4).
Proof. We prove the statement about the fibers of (4.5), the other statement being analogous.
Let (E,Ψ) be a point of the fiber, then the fiber of locx over (E|∆x ,Ψ|∆x) is non-empty. Then
by Lemma 4.11 this fiber is isomorphic to Fibd+e(X,x), where e is the degree of (E|∆x ,Ψ|∆x)
(recall that e ≥ 0). Since Fibd+e(X,x) classifies nonpositive vector bundles, we get d + e ≤ 0.
Thus N > N(e, λ) and we see that (F,Φ) is stabilized. �
4.3.5 Proof of Proposition 4.10
Let us take an integer N larger than N(j, λ) for all j = 0, . . . ,−d. It is enough to show that the
motivic functions
A :=
∑
d′+d′′=d
[
Pairnilp,−
r,r•,d′
(X,x, λ)× Pairnilp,−
r,d′′
(
P1,∅, λ
)
→ Pairloc,fl
r•,N
× Pairloc
N
]
and
B :=
∑
d′+d′′=d
[
Pairnilp,−
r,r•,d′
(
P1,∞, λ
)
× Pairnilp,−
r,d′′ (X,∅, λ)→ Pairloc,fl
r•,N
× Pairloc
N
]
are equal. The morphisms are locfl
x,N × loc∞,N and locfl
∞,N × locx,N respectively.
Let K ⊃ k be an extension and let ξ be a K-point of the stack Pairloc,fl
r•,N
×Pairloc
r,N represented
by ((F,Φ, F•), (F
′,Φ′)). By Proposition 2.1 it is enough to show that ξ∗A = ξ∗B. Using base
change, we may assume that K = k. According to Lemma 4.14, these motivic functions are
zero unless (F,Φ) and (F ′,Φ′) are stabilized.
Let us lift (F,Φ) and (F ′,Φ′) to (Or,Φ) and (Or,Φ
′
), where Φ′,Φ
′ ∈ glr,O. Choose kernel-
strict g, g′ ∈ GLr,K such that g, g−1, g′, and (g′)−1 have poles of order less than N/2 and such
that gΦg−1 = g′Φ
′
(g′)−1 = nλ.
Then, according to Lemmas 4.12 and 4.13 (applied to X and P1) we get
ξ∗A =
∑
d′+d′′=d
[Fibd′+e(X,x)][Fibd′′+e′(P1,∞)]
[Zg][Zg′ ]
=
∑
d′+d′′=d+e+e′
[Fibd′(X,x)][Fibd′′(P1,∞)]
[Zg][Zg′ ]
.
Similarly,
ξ∗B =
∑
d′+d′′=d
[Fibd′+e(P1,∞)][Fibd′′+e′(X,x)]
[Zg][Zg′ ]
=
∑
d′+d′′=d+e+e′
[Fibd′(P1,∞)][Fibd′′(X,x)]
[Zg][Zg′ ]
.
We see that ξ∗A = ξ∗B. This completes the proof of Proposition 4.10 and thus the proof of
Theorem 4.4. �
26 R. Fedorov, A. Soibelman and Y. Soibelman
Remark 4.15. We emphasize that we only worked with motivic classes of Artin stacks of finite
type. It seems plausible that one can define motivic classes of stacks like
[
Fibd(X,x)
]
using some
ideas of motivic integration. This would significantly simplify our argument. Unfortunately, we
were not able to develop such a formalism.
4.4 Case of P1 and two points
Consider the case when X = P1, D = {0,∞}.
Proposition 4.16. We have in Mot(k)[[Γ′+]] ⊂ Mot(k)
[[
w,w0,•, w∞,•, z
−1
]]
[
Pairnilp,−(P1, {0,∞}
)]
= Exp
(
w
∑∞
j=1
∑∞
j′=1w0,jw∞,j′
(L− 1)(1− z−1)
)
, (4.6)
where Exp is the plethystic exponent defined in Section 3.4.
Proof. We note that the proof in [30, Section 5.4] goes through in the motivic case as well.
The only difference is that Mellit uses the Hall algebra of the Jordan quiver (that is, the Hall
algebra of the category of vector spaces with nilpotent endomorphisms); this Hall algebra has
to be replaced with the similar motivic Hall algebra in our case.
Let us give more details. In [12, Section 5], to any smooth projective geometrically connected
curve over k we associated the Hall algebra of the category of coherent sheaves on this curve,
denoted by H. Let us take the curve to be P1
k and let H0 be the subalgebra of torsion sheaves
supported at 0 ∈ P1
k. Since such sheaves are identified with finite dimensional representations of
the Jordan quiver, we can view H0 as the Hall algebra of the category of such representations.
Of course, we could have taken any other curve and any rational point.
Following Mellit, we re-write the LHS of (4.6) in terms of products of certain elements of this
algebra. This part of Mellit’s proof is geometric, so it is easily carried to the motivic case. The
rest of the proof is a calculation in this Hall algebra; the necessary identities in the Hall algebra
are easily derived from results of [12, Section 5]. �
Remark 4.17. Note that the RHS of (4.6) can be written without plethystic exponents as
follows (cf. [12, Lemma 5.7.3])
−∞∏
d=0
∞∏
j=1
∞∏
j′=1
1 +
∑
i≥1
Li(i−1)(
Li − 1
)
· · ·
(
Li − Li−1
)zidwiwi0,jwi∞,j′
.
4.5 Motivic modified Macdonald polynomials
For a commutative unital ring A, let SymA[w•] be the ring of symmetric functions with coef-
ficients in A in variables w•. In this section we define axiomatically the images of modified
Macdonald polynomials in SymMot(k)[[z]][w•].
Consider the modified Macdonald polynomials H̃λ(w•; q, z) ∈ SymZ[q,z][w•]. For a definition
see, for example, [30, Definition 2.5]. It is not clear from this definition that the coefficients
of H̃λ(w•; q, z) are integers, but this is well-known (see, e.g., [14] and references therein). Note
that H̃λ is a symmetric function so, formally speaking, it is not a polynomial.
We denote by H̃mot
λ (w•; z) ∈ SymMot(k)[z][w•] the image of the corresponding modified Mac-
donald polynomial under the homomorphism SymZ[q,z][w•] → SymMot(k)[z][w•] sending q to L;
we call these images motivic modified Macdonald polynomials.
Define the motivic Hall–Littlewood polynomials as the specialization
Hmot
λ (w•) := H̃mot
λ (w•; 0).
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 27
Thus Hmot
λ is the image of the usual Hall–Littlewood polynomial under the homomorphism
Z[q, w•] → Mot(k)[w•] sending q to L. The motivic Hall–Littlewood polynomials can be inter-
preted as follows: let Flλ stand for the scheme of all flags in k|λ| preserved by nλ. Then Flλ is
graded by the type of the flag. It is not difficult to check that we have
Hmot
λ (w•) = [Flλ] ∈ Mot(k)[w•]
(cf. [30, Theorem 2.12, Corollary 2.13]).
It follows from [28, Chapter 3, equation (2.7)] that Hmot
λ form a basis of the Mot(k)-module
SymMot(k)[w•]. Thus H̃mot
λ also form a basis of SymMot(k)[[z]][w•].
Proposition 4.18.
(a) The motivic modified Macdonald polynomials H̃mot
λ satisfy the following properties
(i)
Exp
∞∑
j=1
∞∑
j′=1
w0,jw∞,j′
(L− 1)(1− z)
=
∑
λ
aλ(z)H̃mot
λ (w0,•; z)H̃
mot
λ (w∞,•; z), (4.7)
where aλ are invertible elements of Mot(k)[[z]].
(ii) H̃mot
λ =
∑
µ : µ′≺λ′
bλµH
mot
µ , where bλµ are some elements of Mot(k)[[z]] such that bλλ
are invertible. (Here µ′ and λ′ stand for the conjugate partitions, ≺ is the usual order
on partitions.)
(iii) H̃mot
λ (1•; z) = 1, where 1• stands for the sequence (1, 0, . . . , 0, . . . ).
(iv) H̃mot
λ is homogeneous in w• of degree |λ|.
(b) The motivic modified Macdonald polynomials are uniquely determined by properties (i)–
(iv).
(c) Additionally we have
aλ(z) =
1∏
h∈Hook(λ)
(
La(h) − zl(h)+1
)(
La(h)+1 − zl(h)
) , (4.8)
where Hook(λ) stands for the set of hooks of λ, a(h) and l(h) stand for the armlength and
the leglength of the hook h respectively.
Proof. (a) It is enough to check the corresponding properties for the usual modified Macdonald
polynomials H̃λ ∈ SymZ[q,z][w•] and Hall–Littlewood polynomials Hλ ∈ SymZ[q][w•]. To prove
property (ii) we note first that according to [30, Definition 2.5] we have
H̃λ[(q − 1)w•] =
∑
µ : µ′≺λ′
cλµ(q, z)mµ′(w•),
where [(q − 1)w•] stands for the plethystic action as in [30, Section 2.1]. Recalling that the
Hall–Littlewood polynomials are z = 0 specializations of the modified Macdonald polynomials,
we get
Hλ[(q − 1)w•] =
∑
µ : µ′≺λ′
cλµ(q, 0)mµ′(w•).
28 R. Fedorov, A. Soibelman and Y. Soibelman
Now it is easy to see that an analogue of property (ii) holds in SymQ[q,z](w•): we can write H̃λ =∑
µ : µ′≺λ′
b′λµHµ, where b′λµ are some elements of Q[q, z]. Next, Hλ form a basis in SymZ[q](w•)
(by [28, Chapter 3, equation (2.7)]), so H̃λ form a basis in SymZ[q][[w]](w•). It follows that
b′λµ ∈ Z[w][[z]] and b′λλ are invertible in this ring. Now property (ii) follows.
It is sufficient to prove properties (iii), and (iv) in SymQ(q,z)[w•]. Property (iii) is clear
from [30, Definition 2.5]. Property (iv) follows, for example, from the definition of H̃ given
in [13].
We first prove an analogue of property (i) in SymQ(q,z)[w•]. Recall from loc. cit. that
SymQ(q,z)[w•] carries the q, z-scalar product (·, ·)q,z. for which
Exp
∞∑
j=1
∞∑
j′=1
w0,jw∞,j′
(q − 1)(1− z)
is the reproducing kernel. This means that if fλ(w•; q, z) is any graded Q(q, z)-basis in
SymQ(q,z)[w•] indexed by partitions and f∨λ (w•; q, z) is the dual basis with respect to (·, ·)q,z,
then
Exp
∞∑
j=1
∞∑
j′=1
w0,jw∞,j′
(q − 1)(1− z)
=
∑
λ
fλ(w0,•; q, z)f
∨
λ (w∞,•; q, z). (4.9)
Next, by property (iv) the basis Hλ(w•; q, z) is a graded basis. Thus, by [30, Proposition 2.7] the
dual of Hλ(w•; q, z) is equal to aλ(q, z)Hλ(w•; q, z) for some aλ(q, z) ∈ Q(q, z). Further, in [30,
Section 2.4] it is shown that
aλ(q, z) =
1∏
h∈Hook(λ)
(
qa(h) − zl(h)+1
)(
qa(h)+1 − zl(h)
) . (4.10)
It is clear from this formula that aλ(q, z) ∈ Z(q)[[z]]. Now property (i) follows from (4.9).
The condition in part (c) follows from (4.10).
Now we prove part (b). Since H̃mot
λ form a basis of SymMot(k)[[z]][w•], there is a unique
Mot(k)[[z]]-linear scalar product on SymMot(k)[[z]][w•] such that 〈H̃mot
λ , H̃mot
µ 〉 = δλµ
1
aλ
. (This is
the scalar product such that the LHS of (4.7) is the reproducing kernel for this product).
Let H ′λ = H ′λ(w•; z) be symmetric functions satisfying conditions of part (a), we need to show
that H ′λ = H̃mot
λ . Applying condition (ii), we see that we can write
H ′λ =
∑
µ : µ′≺λ′
cλµH̃
mot
µ , (4.11)
where cλλ is invertible. Condition (i) shows that we have∑
λ
aλ(z)H̃mot
λ (w0,•; z)H̃
mot
λ (w∞,•; z) =
∑
λ
a′λ(z)H ′λ(w0,•; z)H
′
λ(w∞,•; z),
with invertible a′λ(z). Recalling that H ′λ form a graded basis in SymMot(k)[[z]][w•], we see that
〈H ′λ, H ′µ〉 = δλµ
1
a′λ
. Indeed, Hmot
λ and aλH
mot
λ are dual basis for the scalar product, so the above
equality shows that H ′λ and a′λH
′
λ are dual basis as well.
We prove that H ′λ = H̃mot
λ by induction on the conjugate partition λ′ with respect to ≺.
Thus we assume that H ′µ = H̃mot
µ whenever µ′ ≺ λ′. Taking the scalar product of (4.11) with
H̃mot
µ = H ′µ, we see that cλµ〈H̃mot
µ , H̃mot
µ 〉 = 0 whenever µ 6= λ. We see that cλµ = 0 so that
H ′λ = cλλH̃
mot
λ . Now condition (iii) implies that cλλ = 1. �
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 29
4.6 Explicit formulas for the graded motivic classes of nilpotent pairs
Now we are ready to give the precise formula for
[
Pairnilp,−(X,D, λ)
]
. Recall that for a par-
tition λ we defined Jmot
λ (z), Hmot
λ (z) ∈ Mot(k)[[z]] in [12, Section 1.3.2]. In this paper, we
will denote them by Jmot
λ,X (z) and Hmot
λ,X (z) respectively to emphasize that they depend on the
curve X and to ensure that they are not confused with motivic modified Macdonald polynomials
H̃mot
λ (w•; z) and with motivic Hall–Littlewood polynomials Hmot
λ (w•). Denote by g the genus
of X.
Theorem 4.19. We have in Mot(k)[[Γ′+]][
Pairnilp,−(X,D, λ)
]
= w|λ|L(g−1)〈λ,λ〉Jmot
λ,X
(
z−1
)
Hmot
λ,X
(
z−1
) ∏
x∈D
H̃mot
λ
(
wx,•; z
−1
)
.
Proof. According to [12, Theorem 1.4.1], we have∑
λ
[
Pairnilp,+(X,∅, λ)
]
= w|λ|
∑
λ
L(g−1)〈λ,λ〉Jmot
λ,X (z)Hmot
λ,X (z),
where the superscript “+” stands for HN-nonnegative vector bundles (see Section 3.2 and [12,
Section 3.2] for the definition). Inspecting the proof, we see that for each λ the summands are
equal:[
Pairnilp,+(X,∅, λ)
]
= w|λ|L(g−1)〈λ,λ〉Jmot
λ,X (z)Hmot
λ,X (z).
By Lemma 3.2 we get an isomorphism of stacks
Pairnilp,+
r,d (X,∅, λ) ' Pairnilp,−
r,−d (X,∅, λ).
Thus [
Pairnilp,−(X,∅, λ)
]
= w|λ|L(g−1)〈λ,λ〉Jmot
λ,X
(
z−1
)
Hmot
λ,X
(
z−1
)
.
To be able to apply Theorem 4.4, we need the following lemma.
Lemma 4.20. We have in Mot(k)[[w∞,•, z
−1]][
Pairnilp,−(P1,∞, λ
)][
Pairnilp,−(P1,∅, λ
)] = H̃mot
λ
(
w∞,•; z
−1
)
.
Proof. Our proof is similar to that of [30, Theorem 5.5]. Let H ′λ ∈ Mot(k)[[w•, z]] be the series
such that[
Pairnilp,−(P1,∞, λ
)][
Pairnilp,−(P1,∅, λ
)] = H ′λ
(
w∞,•; z
−1
)
.
Denote by Cλµ the stack classifying pairs (E,Ψ), where E is a nonpositive vector bundle of
rank |λ| on P1, Ψ is an endomorphism of E generically conjugate to nλ and conjugate to nµ at
x = ∞. Then Cλµ is graded by the degree of E, and we have [Cλµ] ∈ Mot(k)
[[
z−1
]]
. Clearly,
we have
H ′λ
(
w•; z
−1
)
=
∑
µ
[Cλµ][
Pairnilp,−(P1,∅, λ
)]Hmot
µ (w•), (4.12)
where Hmot
µ are the motivic Hall–Littlewood polynomials. Note that Cλµ = ∅ unless µ′ ≺ λ′
because for all i the dimension of the fiber of Ker Ψi is semicontinuous on P1. Now it is easy to see
30 R. Fedorov, A. Soibelman and Y. Soibelman
that H ′λ are symmetric functions with coefficients in Mot(k)[[z]]. We will use Proposition 4.18(b)
to show that for all λ we have H ′λ = H̃mot
λ .
To show that H ′λ satisfy property (ii) of Proposition 4.18(a) it remains to show that [Cλλ] is
invertible. This is completely similar to the proof of Lemma 4.3.
To show that H ′λ satisfy condition (i) of Proposition 4.18(a), we note that combining Theo-
rem 4.4 and Proposition 4.16, we get
Exp
∞∑
j=1
w0,jw∞,j
(L− 1)(1− z)
=
∑
λ
a′λ(z)H ′λ(w0,•; z)H
′
λ(w∞,•; z),
where a′λ(z−1) =
[
Pairnilp,−(P1,∅, λ
)]
and the statement follows.
Condition (iii) of Proposition 4.18(a) is obvious. Condition (iv) follows from (4.12). We note
also for the future use that it is clear from the argument that we have[
Pairnilp,−(P1,∅, λ
)]
= a′λ
(
z−1
)
= aλ
(
z−1
)
, (4.13)
where aλ(z) is given by (4.8). The proof of Lemma 4.20 is complete. �
Now Theorem 4.4 completes the proof of Theorem 4.19. �
Corollary 4.21. We have in Mot(k)[[Γ′+]]
[
Pair−(X,D)
]
= Pow
(∑
λ
w|λ|L(g−1)〈λ,λ〉Jmot
λ,X
(
z−1
)
Hmot
λ,X
(
z−1
) ∏
x∈D
H̃mot
λ
(
wx,•; z
−1
)
,L
)
.
Proof. The argument is similar to the proof of [12, Proposition 3.8.1]. In more details, let
(E,Ψ, E•,•) be a K-point of Pair−(X,D). According to [12, Lemma 3.8.3], we can uniquely
decompose
(E,Ψ)
'−→
⊕
i
Rk(xi)/K(Ei, xi Id +Ψi),
where xi are distinct closed points of A1
K (the eigenvalues of Ψ), (Ei,Ψi) are k(xi)-points of the
stack Pairnilp,−(X,∅), k(xi) ⊃ K is the residue field of xi, and Rk(xi)/K is the pushforward
functor. It follows easily from the proof of [12, Lemma 3.8.3], that we can write uniquely
(E,Ψ, E•,•)
'−→
⊕
i
Rk(xi)/K(Ei, xi Id +Ψi, Ei,•,•),
where (Ei,Ψi, Ei,•,•) are k(xi)-points of Pairnilp,−(X,D). It remains to use a version of [12,
Lemma 3.8.2]. �
4.7 Case of P1
In the case of X = P1 we can give a more explicit answer. Moreover, we get an answer valid
in Mot(k)[[Γ′+]] rather than in its completion Mot(k)[[Γ′+]], which is desirable, since we do not
know whether the natural homomorphism Mot(k) → Mot(k) is injective. We argue as in [30,
Corollary 5.9]. Combining Theorem 4.4, (4.13), and (4.8), we get the following formula valid in
Mot(k)[[Γ′+]].
[
Pairnilp,−(P1, D, λ
)]
= w|λ|
∏
x∈D
H̃mot
λ
(
wx,•; z
−1
)
∏
h∈Hook(λ)
(
La(h) − z−l(h)−1
)(
La(h)+1 − z−l(h)
) ,
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 31
where Hook(λ) stands for the set of hooks of λ, a(h) and l(h) stand for the armlength and the
leglength of the hook h respectively. Arguing as in Corollary 4.21, we get in Mot(k)[[Γ′+]]
[
Pair−
(
P1, D
)]
= Pow
∑
λ
w|λ|
∏
x∈D
H̃mot
λ (wx,•; z)∏
h∈Hook(λ)
(
La(h) − z−l(h)−1
)(
La(h)+1 − z−l(h)
) ,L
. (4.14)
5 Parabolic Higgs bundles with fixed eigenvalues
5.1 Stacks of Higgs bundles
Let X and D be as above. From now on we denote by g the genus of X. Our goal in this
section is to calculate the motivic Donaldson–Thomas series of the category of parabolic Higgs
bundles. More precisely, we calculate the motivic classes of the moduli stacks of Higgs bundles
with fixed eigenvalues and with nonpositive underlying vector bundles. Our argument is similar
to [12, Sections 3.4–3.5]. The main result is Corollary 5.9. We denote by ΩX the canonical line
bundle on X.
Definition 5.1. A parabolic Higgs bundle of type (X,D) is a triple (E,E•,•,Φ), where (E,E•,•)
is a point of Bunpar(X,D), Φ: E → E ⊗ΩX(D) is an OX -linear morphism (called a Higgs field
on (E,E•,•)) such that for all x ∈ D and j ≥ 0 we have Φx(Ex,j) ⊂ Ex,j ⊗ ΩX(D)x.
We denote the category (and the Artin stack) of parabolic Higgs bundles by Higgs =
Higgs(X,D). We define the Γ′+-graded stack Higgs− = Higgs−(X,D) following the general
formalism of Section 3.3, that is, Higgs− is the open substack of Higgs corresponding to Higgs
bundles with nonpositive underlying vector bundle. Clearly, this stack is of finite type in the
graded sense (that is, the graded components are of finite type). In this section X and D are
fixed, so we skip them from the notation.
5.2 Existence of Higgs fields with prescribed residues
To determine a criterion for the existence of a Higgs bundle with prescribed residues, we use an
approach similar to [2, 32, 33]. Let E → X be a vector bundle and let Φ: E → E ⊗ ΩX(D) be
a morphism. In this case, for all x ∈ D we have a residue Resx Φ ∈ End(Ex).
Proposition 5.2. Let E be a vector bundle on X and let for x ∈ D, ρx ∈ End(Ex) be an
endomorphism of the fiber of E at x. There exists a Higgs field Φ: E → E ⊗ ΩX(D) with
Resx Φ = ρx for all x ∈ D if and only if∑
x∈D
tr(ρxϕx) = 0
for all ϕ ∈ End(E), where tr stands for the trace.
Proof. Consider the short exact sequence of sheaves
0→ OX → KX → KX/OX → 0,
where KX is the constant sheaf corresponding to the function field of X. Let ΩK be the constant
sheaf of meromorphic differential forms on X. We can obtain a new short exact sequence of
sheaves:
0→ End(E)⊗ ΩX → End(E)⊗ ΩK → End(E)⊗ (ΩK/ΩX)→ 0
32 R. Fedorov, A. Soibelman and Y. Soibelman
by taking the tensor product of the first sequence with End(E)⊗ΩX . Note that the middle term
in this sequence is a constant sheaf, while the last term is an (infinite) direct sum of skyscraper
sheaves. That is, End(E) ⊗ (ΩK/ΩX) ∼=
⊕
x∈X(ix)∗(End(Ex) ⊗ (ΩK/ΩX)x), where (ΩK/ΩX)x
is the vector space of polar parts at x of meromorphic 1-forms, the summation is taken over
all closed points of X, and ix : x → X is the inclusion. Passing to the long exact sequence for
cohomology we obtain the following exact sequence of vector spaces:
End(E)⊗ ΩK →
⊕
x∈X
End(Ex)⊗ (ΩK/ΩX)x → H1(X, End(E)⊗ ΩX)→ 0.
This implies that H1(X, End(E)⊗ ΩX) may be presented as the quotient of⊕
x∈X
End(Ex)⊗ (ΩK/ΩX)x
by the image of End(E)⊗ΩK (compare with the adelic description of cohomology given in [38,
Chapter 2, Section 5]). Further note that the required Higgs field Φ always exists locally, defined
as Φx = ρx
dzx
zx
, where zx is an étale coordinate near x if x ∈ D and Φx = 0 if x /∈ D. Under
the above presentation of H1(X, End(E) ⊗ ΩX), the local solutions Φx define a cohomology
class a(E,D, ρ•) ∈ H1(X, End(E) ⊗ ΩX). Moreover, it follows from the exact sequence that
a(E,D, ρ•) = 0 if and only if Φ can be defined globally.
Serre duality defines a bilinear pairing H1(X, End(E)⊗ΩX)×End(E)→ k. Using the above
presentation for H1(X, End(E)⊗ ΩX) this pairing may be evaluated on ϕ ∈ End(E) as
〈a(E,D, ρ•), ϕ〉 =
∑
x∈X
Resx tr(Φxϕx) =
∑
x∈D
tr(ρxϕx).
Since the pairing is perfect,
∑
x∈D
tr(ρxϕx) = 0 for all ϕ ∈ End(E) if and only if a(E,D, ρ•) = 0.
The proof is complete. �
5.3 Parabolic Higgs bundles with fixed eigenvalues
Recall that k[D×Z>0] is the set of all k-valued sequences ζ = ζ•,• = (ζx,j) indexed by D×Z>0
such that ζx,j = 0 for j � 0.1 For ζ ∈ k[D×Z>0] let Higgs(ζ) = Higgs(X,D, ζ) denote the full
subcategory of Higgs (and its stack of objects) corresponding to collections (E,E•,•,Φ) such
that (Φ− ζx,j1)(Ex,j−1) ⊂ Ex,j ⊗ ΩX(D)x for all x ∈ D and j > 0. Again, the Γ′+-graded stack
Higgs−(ζ) is defined following the formalism of Section 3.3.
Let ζ ∈ k[D × Z>0] and let γ = (r, r•,•, d) ∈ Γ+. Recall from Section 3.6 that we have set
deg0,ζ γ :=
∑
x∈D
∞∑
j=1
ζx,jrx,j ∈ k.
Lemma 5.3. Let E ∈ Bunpar(k) and ζ ∈ k[D × Z>0]. There exists an object (E,Φ) ∈
Higgs(ζ)(k) if and only if deg0,ζ E′ = 0 for any direct summand E′ of E.
Note that, in particular, this condition implies that deg0,ζ E = 0.
Proof. The proof is the same as the proof of [7, Theorem 7.1] after replacing b(E) with 0
and replacing [7, Theorem 7.2] with Proposition 5.2. (The Atiyah class b(E) represents the
obstruction to existence of a connection (without poles) on E. Thus, it is absent for Higgs fields
essentially because every vector bundle possesses the zero Higgs field.) �
1According to our convention we should denote ζ by ζ•,• but it does not look nice in the formulas.
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 33
5.4 Parabolic pairs with isoslopy underlying parabolic bundles
Recall that for κ ∈ k and ζ ∈ k[D × Z>0], a parabolic bundle E ∈ Bunpar is (κ, ζ)-isoslopy if
degκ,ζ E′
rk E′
=
degκ,ζ E
rk E
whenever E′ is a direct summand of E. Similarly to [12, Lemma 3.2.2], one checks that the notion
of (κ, ζ)- isoslopy parabolic bundle is invariant with respect to field extensions. Thus for each
γ ∈ Γ′+ we have a well-defined subset Bunpar,(κ,ζ)−iso,−
γ ⊂ |Bunpar,−
γ |. As in [12, Lemma 3.2.3]
we show that this subset is constructible.
Let Pair(κ,ζ)−iso,−
γ be the preimage of Bunpar,(κ,ζ)−iso,−
γ under the projection |Pair|→|Bunpar|.
Proposition 5.4. For γ = (r, r•,•, d) ∈ Γ+, set χ(γ) := (g − 1)r2 +
∑
x∈D
∑
j<j′
rx,jrx,j′. Then
[Higgs−γ (ζ)] =
{
Lχ(γ)
[
Pair(0,ζ)−iso,−
γ
]
if deg0,ζ γ = 0,
0 otherwise.
Proof. The case of deg0,ζ γ 6= 0 is obvious in view of Lemma 5.3. Assume that deg0,ζ γ = 0. It
is enough to show the equality of motivic functions in Mot
(
Bunpar,−
γ
)
:[
Higgs−γ (ζ)→ Bunpar,−
γ
]
= Lχ(γ)
[
Pair(0,ζ)−iso,−
γ → Bunpar,−
γ
]
. (5.1)
Let K ⊃ k be a field extension. Let ξ : SpecK → Bunpar,−
γ be a point represented by a parabolic
bundle E = (E,E•,•). In view of Proposition 2.1, we only need to check that the ξ-pullbacks
of (5.1) are equal. If E is not (0, ζ)-isoslopy, then, by Lemma 5.3, the pullbacks are equal to
zero, so we assume that E is (0, ζ)-isoslopy.
Let Higgs(E, ζ) denote the space of Higgs fields on E with eigenvalues ζ (that is, the E-fiber
of the projection Higgs(ζ)→ Bunpar). By Lemma 5.3, Higgs(E, ζ) is non-empty, so it is a torsor
over the vector space Higgs(E, 0). Thus,
ξ∗
[
Higgs−γ (ζ)→ Bunpar,−
γ
]
= Ldim Higgs(E,0).
On the other hand, we have
ξ∗
[
Pair(0,ζ)−iso,−
γ → Bunpar,−
γ
]
= Ldim End(E).
It remains to prove the following lemma.
Lemma 5.5. Let E ∈ Bunpar
γ be a parabolic bundle. Then
dim End(E)− dim Higgs(E, 0) = −χ(γ).
Proof. Write E = (E,E•,•). Let End(E) ⊂ End(E) be the subsheaf of endomorphisms pre-
serving flags. One checks easily that the trace pairing gives an isomorphism between the dual
sheaf End(E)∨ ⊗ ΩX and Higgs(E, 0), where Higgs(E, 0) stands for the sheaf of Higgs fields
on E with zero eigenvalues. Thus by Riemann–Roch theorem we have
dim End(E)− dim Higgs(E, 0) = h0(X, End(E))− h0
(
X, End(E)∨ ⊗ ΩX
)
= (1− g) rk End(E) + deg End(E).
It remains to calculate deg End(E). For x ∈ D consider the fiber Ex, its ring of endomorphisms
End(Ex), its subspace Vx of endomorphisms preserving the flag Ex,•, and the quotient of vector
34 R. Fedorov, A. Soibelman and Y. Soibelman
spaces Wx := End(Ex)/Vx. Further, consider the torsion sheaf W := ⊕x∈D(ix)∗Wx, where, as
before, ix : x→ X is the inclusion. We have an exact sequence
0→ End(E)→ End(E)→W → 0,
so
deg End(E) = deg End(E)− length(W ) = 0−
∑
x∈D
∑
j<j′
rx,jrx,j′
and the lemma follows. �
The lemma completes the proof of Proposition 5.4. �
Proposition 5.6. We have in Mot(k)[[Γ′+]]
[Pair−] =
∏
τ∈k
∑
γ∈Γ′+
degκ,ζ γ=τ rk γ
[
Pair(κ,ζ)−iso,−
γ
]
eγ
.
We note that the product makes sense because for a γ ∈ Γ′+ there are only finitely many ways
to write γ as the sum of elements of Γ′+. Also, the order of the multiples is irrelevant, since we
are working with a commutative quantum torus.
Proof. The proof is almost the same as the proof of [12, Lemma 3.5.3 ] (see also [12, Proposi-
tion 3.5.1]). �
We need some notation. Let us write
L · Log
(∑
λ
w|λ|Jmot
λ,X
(
z−1
)
Hmot
λ,X
(
z−1
) ∏
x∈D
H̃mot
λ
(
wx,•; z
−1
))
=
∑
γ∈Γ′+
Bγeγ ,
where Log is the plethystic logarithm defined in Section 3.4, the summation is over all partitions.
We note that Bγ are W -invariant, where W =
∏
x∈D
Σ∞ (cf. Remark 4.2). Note also that B0 = 0
by the definition of plethystic logarithm.
Definition 5.7. The motivic classes Bγ ∈ Mot(k) are called motivic Donaldson–Thomas in-
variants of the pair (X,D).
Corollary 5.8. For each τ ∈ k we have in Mot(k)[[Γ′+]]
∑
γ∈Γ′+
degκ,ζ γ=τ rk γ
[
Pair(κ,ζ)−iso,−
γ
]
eγ = Exp
∑
γ∈Γ′+
degκ,ζ γ=τ rk γ
Bγeγ
,
where Exp is the plethystic exponent defined in Section 3.4.
Proof. First of all, using Corollary 4.21 and properties of plethystic operations, we get
[Pair−] = Exp
∑
γ∈Γ′+
Bγeγ
=
∏
τ∈k
Exp
∑
γ∈Γ′+
degκ,ζ γ=τ rk γ
Bγeγ
.
Now, it remains to use Proposition 5.6 and equate the slopes (cf. [12, Lemma 3.7.1]). �
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 35
Corollary 5.9. We have in Mot(k)[[Γ′+]]
∑
γ∈Γ′+
L−χ(γ)[Higgs−γ (ζ)]eγ = Exp
∑
γ∈Γ′+
deg0,ζ γ=0
Bγeγ
,
where Exp is the plethystic exponent defined in Section 3.4.
Proof. Use Proposition 5.4 and Corollary 5.8. �
5.5 Case of P1
Assume now that X = P1. Then we have a simpler result. Moreover, it is more precise in the
sense that we get an answer in Mot(k) rather than in Mot(k). Define the Donaldson–Thomas
invariants Bγ ∈ Mot(k) by
L · Log
∑
λ
w|λ|
∏
x∈D
H̃mot
λ
(
wx,•; z
−1
)
∏
h∈Hook(λ)
(
La(h) − z−l(h)−1
)(
La(h)+1 − z−l(h)
)
=
∑
γ∈Γ′+
Bγeγ . (5.2)
We have precisely the same formula as in Corollary 5.9, where Bγ is replaced with Bγ (thus, the
formula is valid in Mot(k)). The proof is the same as of Corollary 5.9 except that one uses (4.14)
instead of Corollary 4.21. Comparing Corollary 4.21 with (4.14), we see that the images of Bγ
in Mot(k) are equal to Bγ .
6 Stability conditions for Higgs bundles
6.1 Harder–Narasimhan filtration
Recall that in Section 3.7 we defined the set Stab of sequences of parabolic weights. To every
sequence of parabolic weights we associated a stability condition on parabolic bundles in Defini-
tion 3.8. We want to extend this to Higgs bundles and to calculate the motivic classes of stacks
of semistable parabolic Higgs bundles with nonpositive underlying vector bundles. Let X and D
be as before and let σ ∈ Stab.
Definition 6.1.
(i) A parabolic Higgs bundle (E,Φ) is σ-semistable if (3.6) is satisfied for all strict subbundles
preserved by Φ.
(ii) A parabolic Higgs bundle (E,Φ) = (E,E•,•,Φ) is σ-nonpositive-semistable, if the un-
derlying vector bundle of E is nonpositive and (3.6) is satisfied for all strict subbundles
E′ = (E′, E′•,•) such that Φ preserves E′ and E/E′ is a nonpositive vector bundle.
The notion of σ-nonpositive-semistable Higgs bundle is similar to that of nonnegative-semi-
stable Higgs bundle (see [12, Section 3.3] and [34]). We emphasize that a σ-nonpositive-
semistable parabolic Higgs bundle is not necessarily σ-semistable; cf. [12, Remark 3.3.1].
Denote the substack of Higgs(ζ) corresponding to σ-semistable (resp. σ-nonpositive-semi-
stable) parabolic Higgs bundles by Higgsσ−ss(ζ) (resp. Higgsσ−ss,−(ζ)). An argument similar
to [41, Lemma 3.7] shows that these are open substacks of Higgs(ζ) and Higgs−(ζ) respectively.
Note that if (E,Φ) is a parabolic Higgs bundle and E′ ⊂ E is a strict parabolic subbundle
preserved by Φ, then we get an induced Higgs field on E/E′; denote it Φ′. Then (E/E′,Φ′) ∈
Higgs−(ζ). One can use this construction to give Higgs−(ζ) the structure of an exact category.
The proof of the following proposition is completely similar to the proof of Proposition 3.11.
36 R. Fedorov, A. Soibelman and Y. Soibelman
Proposition 6.2.
(i) If (E,Φ) ∈ Higgs(ζ) is a parabolic Higgs bundle with eigenvalues ζ, then there is a unique
filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Em = E by strict parabolic subbundles preserved by Φ such
that all the quotients Ei/Ei−1 with induced Higgs fields are σ-semistable parabolic Higgs
bundles and we have τ1 > · · · > τm, where τi is the (1, σ)-slope of Ei/Ei−1.
(ii) If (E,Φ) ∈ Higgs−(ζ), then there is a unique filtration of E as in (i) by strict parabolic
subbundles preserved by Φ with quotients being σ-nonpositive-semistable parabolic Higgs
bundles.
6.2 Kontsevich–Soibelman factorization formula
The general formalism of [22] implies the following factorization formula valid in Mot(k)[[Γ′+]].
One can also give a direct proof along the lines of the proof of [12, Proposition 3.6.1]2
∑
γ∈Γ′+
L−χ(γ)[Higgs−γ (ζ)]eγ =
∏
τ∈R
∑
γ∈Γ′+
deg1,σ γ=τ rk γ
L−χ(γ)
[
Higgsσ−ss,−
γ (ζ)
]
eγ
. (6.1)
Now, taking the plethystic logarithms of both sides and using Corollary 5.9, we get the following
statement.
Proposition 6.3. We have in Mot(k)[[Γ′+]]
∑
γ∈Γ′+
deg1,σ γ=τ rk γ
L−χ(γ)
[
Higgsσ−ss,−
γ (ζ)
]
eγ = Exp
∑
γ∈Γ′+
deg0,ζ γ=0
deg1,σ γ=τ rk γ
Bγeγ
.
If X = P1, then the same formula holds in Mot(k)[[Γ′+]] with Bγ replaced by Bγ, where Bγ are
defined by (5.2).
7 Stabilization
7.1 Stabilization of semistable Higgs bundles
Let X and D be as before. We will be assuming that D 6= ∅. Note that this implies that X has
a k-rational divisor of degree one. Set δ := max(2g − 2 + degD, 0). Fix a stability condition
σ ∈ Stab. Our goal in this section is to calculate the motivic class of the moduli stack of σ-
semistable parabolic Higgs bundles without nonnegativity assumption. The main result in this
section is Theorem 7.5. Recall that in the end of Section 3.7 we defined the categories Bunpar,≤τ
and Bunpar,≥τ . These are the full subcategories of Bunpar whose objects are parabolic bundles
with the σ-HN spectrum contained in (−∞, τ ] and [τ,∞) respectively.
We start with the following analogue of [34, Lemma 3.1].
Lemma 7.1. Let E ∈ Bunpar be a parabolic bundle with the σ-HN-spectrum τ1 > τ2 > · · · > τm.
Assume that for some i ∈ {1, . . . ,m− 1} we have τi − τi+1 > δ. Then there are no σ-semistable
Higgs bundles of the form (E,Φ).
2Note that all but countably many multiples are equal to one. We can understand the countable product as
a clockwise product as in [22, 23]. Note, however, that this is a product in a commutative ring.
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 37
Proof. Assume the contrary. We have an exact sequence
0→ E≥ → E→ E≤ → 0,
where E≥ ∈ Bunpar,≥τi , E≤ ∈ Bunpar,≤τi+1 . Then Φ induces a morphism Φ′ : E≥ → E≤⊗ΩX(D).
Note that E≤⊗ΩX(D) ∈ Bunpar,≤τi+1+δ. By Lemma 3.13, Φ′ = 0 and we see that E≥ is preserved
by Φ contradicting σ-semistability of (E,Φ). �
Next, we have an analogue of [34, Lemma 3.2].
Lemma 7.2. Let (E,Φ) be a σ-semistable Higgs bundle. Assume that deg1,σ E < − r(r−1)
2 δ,
where r = rk E. Then E ∈ Bunpar,≤0.
Proof. Let τ1 > τ2 > · · · > τm be the σ-HN-spectrum of E. Denote by ri the jumps of the
ranks of σ-HN-filtration. By Lemma 7.1 we have τi ≥ τ1 − (i− 1)δ. We have
−r − 1
2
δ >
deg1,σ E
r
=
m∑
i=1
τiri
r
≥
m∑
i=1
(τ1 − (i− 1)δ)ri
r
≥ m
r
τ1 −
r − 1
2
δ
and the statement follows. �
Set |σ| :=
∑
x∈D
(supi σx,i − σx,1). We remark that we always have |σ| ≤ degD. We have an
analogue of [34, Corollary 3.3].
Lemma 7.3. Let γ = (r, r•,•, d) ∈ Γ+ be such that d < −r|σ| − r(r−1)
2 δ. Then
Higgsσ−ss
γ (ζ) = Higgsσ−ss,−
γ (ζ).
Proof. For all x and i replace σx,i by σx,i − σx,1. This does not change |σ| and the notion of
semistability but we now have σx,i ≥ 0 for all x and i. Next
deg1,σ γ = d+
∑
x
∞∑
i=1
σx,irx,i ≤ d+
∑
x
(
sup
i
σx,i
)( ∞∑
i=1
rx,i
)
= d+ r|σ|. (7.1)
Let (E,Φ) ∈ Higgsσ−ss
γ (ζ). By (7.1) we have
deg1,σ γ < −
r(r − 1)
2
δ.
By Lemma 7.2 we have E ∈ Bunpar,≤0 ⊂ Bunpar,− (the last inclusion follows from σx,i ≥ 0).
Conversely, assume that (E,Φ) ∈ Higgsσ−ss,−
γ (ζ). Assume for contradiction that (E,Φ) is
not σ-semistable. Then by Proposition 6.2(i) we have an exact sequence 0→ E′ → E→ E′′ → 0
in Bunpar such that Φ preserves E′, and (E′′,Φ′′) is σ-semistable, where Φ′′ is the induced Higgs
field. Using (7.1) we get
deg1,σ E′′
rk E′′
<
deg1,σ E
rk E
≤ d+ r|σ|
r
− r − 1
2
δ ≤ −rk E′′ − 1
2
δ.
Now it follows from Lemma 7.2 that E′′ ∈ Bunpar,≤0.
Write E′′ = (E′′, E′′•,•). Since (E,Φ) is σ-nonpositive-semistable, E′′ cannot be nonposi-
tive. Thus there is E′′′ ⊂ E′′ with degE′′′ > 0. Let E′′′ = (E′′′, E′′′•,•) be the corresponding
parabolic subbundle of E′′. Then deg1,σ E′′′ ≥ degE′′′ > 0, which gives contradiction with
E′′ ∈ Bunpar,≤0. �
38 R. Fedorov, A. Soibelman and Y. Soibelman
Set 1 = (0, 0•,•, 1) ∈ Γ (here 0•,• is the sequence of zeroes indexed by D × Z>0). If γ ∈ Γ+
and γ 6= 0, then γ +N1 ∈ Γ+ for all N ∈ Z.
Corollary 7.4. Let γ = (r, r•,•, d) ∈ Γ+, γ 6= 0, and N > |σ|+ r−1
2 δ + d/r. Then
Higgsσ−ss
γ (ζ) ' Higgsσ−ss,−
γ−Nr1(ζ).
Proof. Since X has a divisor of degree one, it has a line bundle of degree N . Tensorisation with
this line bundle gives Higgsσ−ss
γ (ζ) ' Higgsσ−ss
γ−Nr1(ζ). Now Lemma 7.3 completes the proof. �
Recall from Definition 5.7 the Donaldson–Thomas invariants Bγ ∈ Mot(k). For each τ ∈ R
define the elements Hγ(ζ, σ) ∈ Mot(k), where γ ∈ Γ+ is such that the (1, σ)-slope of γ is τ (or
γ = 0), by the following formula.
∑
γ∈Γ′+
deg0,ζ γ=0
deg1,σ γ=τ rk γ
L−χ(γ)Hγ(ζ, σ)eγ = Exp
∑
γ∈Γ′+
deg0,ζ γ=0
deg1,σ γ=τ rk γ
Bγeγ
. (7.2)
Thus Hγ(ζ, σ) is defined for all γ such that deg0,ζ γ = 0. Note that H0(ζ, σ) = 1. Now we can
formulate our first main result.
Theorem 7.5. Let γ = (r, r•,•, d) ∈ Γ+, γ 6= 0.
(i) The elements Hγ(ζ, σ) are periodic in the following sense: for d < −|σ| − r−1
2 δ we have
Hγ(ζ, σ) = Hγ−r1(ζ, σ).
(ii) The stack Higgsσ−ss
γ (ζ) is of finite type and we have in Mot(k)[
Higgsσ−ss
γ (ζ)
]
= Hγ−Nr1(ζ, σ) (7.3)
whenever N is large enough, provided that deg0,ζ γ = 0 (it suffices to take N > |σ|+ r−1
2 δ+
d/r). If deg0,ζ γ 6= 0, then the stack is empty.
Proof. For part (ii) combine Corollary 7.4 with Proposition 6.3. Part (i) is clear from
part (ii). �
An immediate corollary of the above theorem and formula (7.2) is the following curious
observation.
Corollary 7.6. Assume that we are given γ ∈ Γ+, sets of eigenvalues ζ and ζ ′, and sequences
of parabolic weights σ, σ′. Let τ and τ ′ be (1, σ) and (1, σ′)-slopes of γ respectively. Assume
also that
{γ′ ∈ Γ+ : deg0,ζ γ
′ = 0, deg1,σ γ
′ = τ rk γ} = {γ′ ∈ Γ+ : deg0,ζ′ γ
′ = 0, deg1,σ′ γ
′ = τ ′ rk γ}.
Then we have an equality of motivic classes[
Higgsσ−ss
γ (ζ)
]
=
[
Higgsσ′−ss
γ (ζ ′)
]
.
7.2 Case of P1
If X = P1, we obtain simpler and more precise results. Namely, if we define elements Hγ(ζ, σ)
by the same formula (7.2) but with Bγ instead of Bγ , then (7.3) holds in Mot(k).
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 39
8 Motivic classes of parabolic connections
8.1 Stacks of parabolic connections
Let X and D be as above. Our goal in this section is to calculate the motivic classes of the
moduli stacks of parabolic bundles with connections with prescribed eigenvalues of residues. In
Section 8.3 we put stability conditions on these moduli stacks and calculate the motivic classes
of substacks of semistable parabolic bundles with connections. Our argument is similar to the
argument for Higgs bundles.
Let E be a vector bundle on X. A connection on E with poles bounded by D is a morphism
of sheaves of abelian groups ∇ : E → E ⊗ΩX(D) satisfying Leibniz rule. In this case for x ∈ D
one defines the residue of the connection resx∇ ∈ End(Ex).
Definition 8.1. A parabolic connection of type (X,D) is a triple (E,E•,•,∇), where (E,E•,•)
is a point of Bunpar(X,D), ∇ : E → E ⊗ ΩX(D) is a connection on E such that for all x ∈ D
and j ≥ 0 we have (resx∇)(Ex,j) ⊂ Ex,j .
We denote the category (and the Artin stack) of parabolic connections by Conn=Conn(X,D).
In this section X and D are fixed, so we skip them from the notation.
Recall that k[D × Z>0] is the set of all k-valued sequences ζ = ζ•,• = (ζx,j) indexed by
D × Z>0 such that ζx,j = 0 for j � 0. For ζ ∈ k[D × Z>0] let Conn(ζ) = Conn(X,D, ζ) denote
the full subcategory of Conn (and its stack of objects) corresponding to collections (E,E•,•,∇)
such that (resx∇− ζx,j1)(Ex,j−1) ⊂ Ex,j for all x ∈ D and j > 0. We call the points of Conn(ζ)
the parabolic bundles with connections with eigenvalues ζ.
Let ζ ∈ k[D × Z>0] and let γ = (r, r•,•, d) ∈ Γ+. Recall from Section 3.6 that
deg1,ζ γ = d+
∑
x∈D
∞∑
j=1
ζx,jrx,j ∈ k.
The following lemma is [7, Theorem 7.1] if k = C. The proof in the general case is completely
similar.
Lemma 8.2. Let E ∈ Bunpar(k) and ζ ∈ k[D × Z>0]. There exists an object (E,∇) ∈
Conn(ζ)(k) if and only if deg1,ζ E′ = 0 for any direct summand E′ of E.
Note that, in particular, for every (E,∇) ∈ Conn(ζ)(k) we have deg1,ζ E = 0. Recall from
Section 5.4 the notion of (κ, ζ)-isoslopy parabolic bundle and the stacks Pair(κ,ζ)−iso
γ . Recall
also that for γ = (r, r•,•, d) ∈ Γ+, we have set χ(γ) := (g − 1)r2 +
∑
x∈D
∑
j<j′
rx,jrx,j′ .
Proposition 8.3. We have
[Connγ(ζ)] =
{
Lχ(γ)
[
Pair(1,ζ)−iso
γ
]
if deg1,ζ γ = 0,
0 otherwise.
Proof. The proof is completely analogous to the proof of Proposition 5.4 with Lemma 5.3
replaced by Lemma 8.2. �
8.2 Stabilization of isoslopy parabolic bundles
As in Section 7 we will be assuming that D 6= ∅. Recall that this implies that X has a k-rational
divisor of degree one. As before, set δ := max(2g − 2 + degD, 0).
Recall that every vector bundle E on X has a unique HN-filtration and the slopes of the
quotients form a sequence called the HN-spectrum of E. We start with the following analogue
of [34, Lemma 4.1].
40 R. Fedorov, A. Soibelman and Y. Soibelman
Lemma 8.4. Let E = (E,E•,•) ∈ Bunpar be a parabolic bundle such that E has HN-spectrum
τ1 > τ2 > · · · > τm. Assume that for some i ∈ {1, . . . ,m− 1} we have τi − τi+1 > δ. Then E is
decomposable.
Proof. One shows that the extensions of a parabolic bundle E′′ by a parabolic bundle E′ (in the
sense of Section 3.5) are classified by a vector space Ext1(E′′,E′) dual to Hom(E′,E′′(ΩX(D)).
Let 0 = E0 ⊂ E1 ⊂ · · · ⊂ Em = E be the Harder–Narasimhan filtration of E. Let Ei be the
strict parabolic subbundle with the underlying vector bundle Ei. We have an exact sequence
0→ Ei → E→ E/Ei → 0. Note that by the assumption the Harder–Narasimhan spectrum of Ei
is contained in [τi,∞), while the Harder–Narasimhan spectrum of (E/Ei)(ΩX(D)) is contained
in (−∞, τi). It follows that Hom
(
Ei, (E/Ei)(ΩX(D))
)
= 0. Thus
Ext1(E/Ei,Ei) = Hom
(
Ei, (E/Ei)(ΩX(D))
)∨
= 0.
Thus E ' Ei ⊕ (E/Ei) is decomposable. �
Next, we have an analogue of [34, Corollary 4.2] whose proof is similar to loc. cit. and to that
of Lemma 7.2.
Lemma 8.5. Let E ∈ Bunpar be indecomposable and cl(E) = (r, r•,•, d). Assume that d <
− r(r−1)
2 δ. Then E ∈ Bunpar,−.
Proof. Write E = (E,E•,•). Let τ1 > τ2 > · · · > τm be the HN-spectrum of E. Denote by ri
the jumps of the ranks of HN-filtration. By Lemma 7.1 we have τi ≥ τ1 − (i− 1)δ. We have
−r − 1
2
δ >
d
r
=
m∑
i=1
τiri
r
≥
m∑
i=1
(τ1 − (i− 1)δ)ri
r
≥ m
r
τ1 −
r − 1
2
δ
and the statement follows. �
Fix ζ ∈ k[D × Z>0]. Let | • | be any norm on the Q-vector subspace of k generated by the
components of ζ. If k is embedded into C, we can take the usual absolute value for | • |. We set
|ζ| :=
∑
x∈D
(maxi |ζx,i|). We have an analogue of [12, Lemma 3.2.3(i)] (cf. also Lemma 7.3).
Lemma 8.6. Let γ = (r, r•,•, d) ∈ Γ+ be such that d < −2r|ζ| − r(r−1)
2 δ. Then
Bunpar,(1,ζ)−iso
γ = Bunpar,(1,ζ)−iso,−
γ .
Proof. Take E = (E,E•,•) ∈ Bunpar,(1,ζ)−iso
γ . Assume for a contradiction that E /∈ Bunpar,−.
By Lemma 8.5, E is decomposable. Let E′ be an indecomposable summand of E such that
E′ /∈ Bunpar,−. By the definition of isoslopy bundles, we have
deg1,ζ E
′
rkE′ =
deg1,ζ E
r . Write cl(E′) =
(r′, r′•,•, d
′). We have
d′
r′
≤
deg1,ζ E′
rk E′
+ |ζ| =
deg1,ζ E
r
+ |ζ| ≤ d
r
+ 2|ζ| < −(r − 1)
2
δ ≤ −(r′ − 1)
2
δ
and Lemma 8.5 gives a contradiction. �
Recall that we have 1 = (0, 0•,•, 1) ∈ Γ.
Corollary 8.7. Let γ = (r, r•,•, d) ∈ Γ+, γ 6= 0, and N > 2|ζ|+ r−1
2 δ+d/r. Then Pair(1,ζ)−iso
γ '
Pair(1,ζ)−iso,−
γ−Nr1 . In particular, Pair(1,ζ)−iso
γ is a constructible subset of Pairγ of finite type.
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 41
Proof. Since X has a divisor of degree one, it has a line bundle of degree N . Tensorisation with
this line bundle gives Pair(1,ζ)−iso
γ ' Pair(1,ζ)−iso
γ−Nr1 . Now Lemma 8.6 completes the proof. �
Define the elements Cγ(ζ) ∈ Mot(k), where γ ranges over Γ+ by the following formula
(cf. (7.2))
∑
γ∈Γ′+
deg1,ζ γ=τ rk γ
L−χ(γ)Cγ(ζ)eγ = Exp
∑
γ∈Γ′+
deg1,ζ γ=τ rk γ
Bγeγ
. (8.1)
Now we can formulate our second main result. Recall that Conn(ζ) = ∅ unless deg1,ζ γ = 0.
Theorem 8.8. Let γ = (r, r•,•, d) ∈ Γ+, γ 6= 0 and deg1,ζ γ = 0. Let ζ be an element of
k[D × Z>0] and let | • | be a norm on the Q-vector subspace of k generated by the components
of ζ. Then
(i) The elements Cγ(ζ) are periodic in the following sense: for d < −2|ζ| − r−1
2 δ we have
Cγ(ζ) = Cγ−r1(ζ).
(ii) The stack Connγ(ζ) is of finite type and we have
[Connγ(ζ)] = Cγ−Nr1(ζ),
whenever N is large enough (it suffices to take N > 2|ζ|+ r−1
2 δ + d/r).
Proof. Combine Proposition 8.3, Corollary 8.7, and Corollary 5.8. �
8.2.1 Case of P1
If X = P1, we obtain simpler and more precise results. Define Bγ ∈ Mot(k) by (5.2). Define
Cγ(ζ) ∈ Mot(k) by the same formula (8.1) but with Bγ replaced by Bγ . Then Theorem 8.8
holds in Mot(k).
8.3 Stability conditions for bundles with connections
Recall that in Definition 3.8 we defined the notion of a sequence of parabolic weights. For non-
resonant connections one can work with more general sequences of parabolic weights. Let us
give the definitions.
Definition 8.9. We say that ζ = ζ•,• ∈ k[D × Z>0] is non-resonant if for all x ∈ X and all
i, j > 0 we have ζx,i − ζx,j /∈ Z6=0.
The importance of this definition is in the following lemma.
Lemma 8.10. Let ζ ∈ k[D×Z>0] be non-resonant and let ϕ be a morphism in Conn(ζ) such that
the underlying morphism of vector bundles is generically an isomorphism. Then the underlying
morphism of vector bundles is an isomorphism.
Proof. One easily reduces to the case k = C. Take x ∈ D. Since ζ is non-resonant, one can
find a subset Ω of C containing {ζx,j |j > 0} and such that the exponential function induces
a bijection between Ω and C − 0. Then it is well-known that every regular connection on the
punctured formal disc has a unique extension to the puncture such that the eigenvalues of the
residues are in Ω. The statement follows. �
42 R. Fedorov, A. Soibelman and Y. Soibelman
Define the space of extended sequences of parabolic weights Stab′ as the set of pairs (κ, σ),
where κ ∈ R≥0, σ = σ•,• is a sequence of real numbers, indexed by D × Z>0, such that for all
x ∈ D we have (3.5).
Definition 8.11. Let (κ, σ) ∈ Stab′. A parabolic connection (E,∇) is (κ, σ)-semistable if for
all strict parabolic subbundles E′ ⊂ E preserved by ∇ we have
degκ,σ E′
rk E′
≤
degκ,σ E
rk E
.
We denote by Conn(κ,σ)−ss(ζ) the substack of Conn(ζ) corresponding to (κ, σ)-semistable
connections. An argument similar to [41, Lemma 3.7] shows that this is an open substacks of
Conn(ζ).
Proposition 8.12. Assume that (κ, σ) ∈ Stab′. Assume also that either ζ is non-resonant,
or κ = 1, σ ∈ Stab. If (E,∇) ∈ Conn(ζ), then there is a unique filtration 0 = E0 ⊂ E1 ⊂
· · · ⊂ Em = E by strict parabolic subbundles preserved by ∇ such that all the quotients Ei/Ei−1
with induced connections are (κ, σ)-semistable parabolic bundles with connections and we have
τ1 > · · · > τm, where τi is the (κ, σ)-slope of Ei/Ei−1.
Proof. In the non-resonant case the proof is completely analogous to the proof of [16, Sec-
tion 1.3] in view of Lemma 3.7 and the following lemma.
Lemma 8.13. Let ζ be non-resonant and let E → F be a morphism in Conn(ζ), which is
generically an isomorphism. Then degκ,σ E ≤ degκ,σ F.
Proof. Write E = (E,E•,•) and F = (F, F•,•). Let ϕ : E → F be the underlying morphism of
vector bundles. By Lemma 8.10, ϕ is an isomorphism. Thus dimEx,j ≤ dimFx,j for all x and j.
Therefore
degκ,σ E = κ degE +
∑
x,j>0
σx,j(dimEx,j−1 − dimEx,j)
= κ degE +
∑
x∈D
(
σx,1 rkE +
∑
i>0
(σx,i+1 − σx,i) dimEx,i
)
≤ κ degF +
∑
x∈D
(
σx,1 rkF +
∑
i>0
(σx,i+1 − σx,i) dimFx,i
)
= degκ,σ F. �
In the resonant case, the proof is completely analogous to the proof of [16, Section 1.3] in
view of Lemma 3.12 (cf. Propositions 3.11 and 6.2). �
Remark 8.14. More generally, If ζ is resonant, one can work with any (κ, σ) ∈ Stab′ such that
σx,j−σx,1 ≤ κ for all x and j. However, the notion of stability does not change if we scale (κ, σ).
Thus, we can always assume that κ = 1, in which case σ ∈ Stab, or κ = 0, in which case σ = 0.
The latter case corresponds to the trivial stability condition; the corresponding motivic class
has been calculated in Theorem 8.8.
Similarly to Proposition 6.3 the Kontsevich–Soibelman factorization formula implies the fol-
lowing proposition.
Proposition 8.15. Let (κ, σ) ∈ Stab′. Assume that either ζ is non-resonant or κ = 1 and
σ ∈ Stab. Then we have in Mot(k)[[Γ+]]
∑
γ∈Γ+
L−χ(γ)[Connγ(ζ)]eγ =
∏
τ∈R
∑
γ∈Γ+
degκ,σ γ=τ rk γ
L−χ(γ)
[
Conn(κ,σ)−ss
γ (ζ)
]
eγ
. (8.2)
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 43
Define the elements Cγ(ζ,κ, σ) ∈ Mot(k), where γ ranges over Γ+ by the following formula
(cf. (7.2)) valid for any τ ∈ k, τ ′ ∈ R
∑
γ∈Γ+
deg1,ζ γ=τ rk γ
degκ,σ γ=τ ′ rk γ
L−χ(γ)Cγ(ζ,κ, σ)eγ = Exp
∑
γ∈Γ+
deg1,ζ γ=τ rk γ
degκ,σ γ=τ ′ rk γ
Bγeγ
. (8.3)
Note that we have Cγ(ζ, 0, 0) = Cγ(ζ), where Cγ(ζ) are defined in (8.1). Now we can formulate
our third main result.
Theorem 8.16. Assume that γ ∈ Γ+, γ 6= 0, and deg1,ζ γ = 0. Then for N > 3|ζ|+(rk γ−1)δ/2
we have[
Conn(κ,σ)−ss
γ (ζ)
]
= Cγ−Nr1(ζ,κ, σ).
We note that if κ = 0, and σ = 0, then this theorem is essentially Theorem 8.8.
Proof. We defined Cγ(ζ,κ, σ) when γ ∈ Γ′+. It will be convenient for us to extend this no-
tation to the case when γ is a multiple of 1 by setting Cn1(ζ,κ, σ) = 1. We will prove that[
Conn(κ,σ)−ss
β (ζ)
]
= Cβ−Nr1(ζ,κ, σ) whenever β ∈ Γ+, deg1,ζ β = 0, and N > 3|ζ|+(rkβ−1)δ/2
by induction on rkβ. The base case of rkβ = 0 is obvious. Note that replacing γ with
γ − N(rk γ)1 shifts the (κ, σ)-slopes by κN . Consider the product in the RHS of (8.2) and
∏
τ∈R
∑
γ∈Γ+
degκ,σ γ=τ rk γ
deg1,ζ γ=N rk γ
L−χ(γ)[Cγ−N rk γ1(ζ,κ, σ)]eγ
. (8.4)
We note that this product makes sense because we set C−N rk γ1(ζ,κ, σ) = 1 in the beginning
of the proof. Note also that deg1,ζ β = 0 implies that d/r ≤ |ζ|, where β = (r, r•,•, d), so
N > 2|ζ|+ (r − 1)δ/2 + d/r. Using (8.2) and Theorem 8.8 we easily see that the coefficients of
these products at eβ are both equal to L−χ(β)[Connβ(ζ)]. Expanding the product in the RHS
of (8.2), we see that the coefficient at eβ is equal to
L−χ(β)
[
Conn(κ,σ)−ss
β (ζ)
]
+
∑
β1,...,βn
n∏
i=1
L−χ(βi)
[
Conn(κ,σ)−ss
βi
(ζ)
]
,
where the summation is over all decompositions of β into the sum of n ≥ 2 non-zero elements
of Γ+ such that deg1,ζ βi = 0 and degκ,σ βi = τ rkβi. Similarly, expanding the product (8.4), we
see that the coefficient at eβ is equal to
L−χ(β)[Cβ−N rkβ1(ζ,κ, σ)] +
∑
β1,...,βn
n∏
i=1
L−χ(βi)[Cβi−N rkβi1(ζ,κ, σ)].
It remains to show that the respective terms in the sums are equal. Clearly, we have N >
3|ζ|+ (rkβi − 1)δ/2 and the statement follows from the induction hypothesis. �
As usual, if X = P1 we obtain similar formulas valid in Mot(k) by replacing Bγ with Bγ
in (8.3).
44 R. Fedorov, A. Soibelman and Y. Soibelman
9 Equalities of motivic classes and non-emptiness
of moduli stacks
9.1 Equalities between motivic classes of stacks
In this section we will give a criterion for non-emptiness of stacks Conn(κ,σ)−ss
γ (ζ) orHiggsσ−ss
γ (ζ).
For the stacks Connγ(ζ) such a criterion follows easily from [7, 8]. We reduce the case, when
stability condition is present, to the case of Connγ(ζ) using some equalities between motivic
classes and Proposition 2.3. We will also discuss a relation with Simpson’s non-abelian Hodge
theory.
It follows from Theorem 8.16 that the motivic class of Conn(κ,σ)−ss
γ (ζ) depends only on
the submonoid of Γ+ given by the equations deg1,ζ γ
′ = 0, degκ,σ γ
′ = τ rk γ′, where τ =
degκ,σ γ/ rk γ. Using this fact, one can give a lot of examples of seemingly unrelated moduli
stacks Conn(κ,σ)−ss
γ (ζ) having the same motivic class. An analogous statement for moduli spaces
of parabolic Higgs bundles is the content of Corollary 7.6 above. Finally, we can get a lot of
equalities between motivic classes of parabolic Higgs bundles and motivic classes of connections.
In the following proposition, we show that motivic classes of the form [Connγ(ζ)] are universal.
This proposition should be compared to Proposition 7.6.
Proposition 9.1. Assume that k is not a finite extension of Q. Let ζ ∈ k[D × Z>0]. Let
γ ∈ Γ+. Let (κ, σ) ∈ Stab′. Assume that ζ is non-resonant or κ = 1 and σ ∈ Stab. Set
τ := degκ,σ γ/ rk γ. Then there is ζ ′ ∈ k[D × Z>0] such that
{γ′ ∈ Γ+ : deg1,ζ γ
′ = 0,degκ,σ γ
′ = τ rk γ′} = {γ′ ∈ Γ+ : deg1,ζ′ γ
′ = 0}. (9.1)
Moreover, we have
[
Conn(κ,σ)−ss
γ (ζ)
]
= [Connγ(ζ ′)].
Proof. Choose x ∈ D and set σ′y,j = σy,j if y 6= x, σ′x,j = σx,j − τ . Since for all (r′, r′•,•, d) ∈ Γ′
we have
∑
j r
′
x,j = r′, we see that degκ,σ γ
′ = τ rk γ if and only if degκ,σ′ γ
′ = 0.
Now, let U be the Q-vector subspace of R generated by 1 and all the numbers σ′y,j , where y
ranges over D and j ranges over positive integers. Similarly, let V be the Q-vector subspace
of k generated by all the components ζy,j of ζ. Since k is not a finite extension of Q, there is
a Q-linear embedding U⊕V → k. Moreover, we may assume that (κ, 1) maps to 1. This follows
from a general fact: if L1 is a finite dimensional Q-vector space, L2 is an infinite dimensional
Q-vector space, v1 and v2 are non-zero vectors of L1 and L2 respectively, then there is a Q-linear
embedding L1 ↪→ L2 such that v1 maps to v2.
Next, (σ′•,•, ζ) is an element of (U⊕V )[D×Z>0] and we define ζ ′ to be its image in k[D×Z>0]
under the embedding. It is clear that we have (9.1). Indeed, the equations deg1,ζ γ
′ = 0,
degκ,σ′ γ
′ = 0 can be written as the following equation in U ⊕ V :
d′(1,κ) +
∑
y∈D
∑
j≥0
(r′y,j−1 − r′y,j)(ζy,,j , σ′y,j) = 0,
where γ′ = (r′, r′•,•, d
′). But the image of this equation in k is exactly deg1,ζ′ γ
′ = 0. Now the
equality of motivic classes follows from Theorems 8.8 and 8.16 (see formulas (8.1) and (8.3)). �
Similarly, we have the following proposition.
Proposition 9.2. Assume that k is not a finite extension of Q. Let ζ ∈ k[D × Z>0]. Let
σ ∈ Stab. Let γ ∈ Γ+. Set τ := deg1,σ γ/ rk γ. Then there is ζ ′ ∈ k[D × Z>0] such that
{γ′ ∈ Γ+ : deg0,ζ γ
′ = 0, deg1,σ γ
′ = τ rk γ′} = {γ′ ∈ Γ+ : deg1,ζ′ γ
′ = 0}. (9.2)
Moreover, we have
[
Higgsσ−ss
γ (ζ)
]
= [Connγ(ζ ′)].
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 45
Proof. The proof is analogous to that of Proposition 9.1 except that one finds an embedding U⊕
V ↪→ k taking (0,κ) to 1 and uses Theorem 7.5 and (7.2) instead of Theorem 8.16 and (8.3). �
Remark 9.3. Many motivic classes of parabolic Higgs bundles and parabolic connections are
equal to classes of the form
[
Higgsσ−ss(0)
]
, cf. [12, Theorem 1.2.1]. However, whether these
classes are universal (that is, whether one can write every motivic class
[
Conn(κ,σ)−ss
γ (ζ)
]
and[
Higgsσ−ss
γ (ζ)
]
in this form) is not clear; the reason is that there are restrictions on σ ∈ Stab.
9.2 Simpson’s non-abelian Hodge theory
Assume now that k = C. Let σ ∈ Stab and ζ ∈ C[D × Z>0]. Let Re ζ and Im ζ denote the real
and imaginary parts of ζ. Assume that either σ−2 Re ζ ∈ Stab or σ+2
√
−1 Im ζ is non-resonant.
In this case, for γ ∈ Γ+ we have moduli stacks
Higgsσ−ss
γ (ζ) and Conn(1,σ−2 Re ζ)−ss
γ
(
σ + 2
√
−1 Im ζ
)
.
Assume that deg1,σ γ = 0. Then, according to the results of [39], the corresponding categories
are equivalent (see especially that table on p. 720 in loc. cit.). Note also that this equivalence
of categories can be upgraded to a diffeomorphism of coarse moduli spaces (cf. [4, 5, 35]).
Proposition 9.4. We have in Mot(C):
[
Higgsσ−ss
γ (ζ)
]
=
[
Conn(1,σ−2 Re ζ)−ss
γ
(
σ+2
√
−1 Im ζ
)]
.
Proof. Note that the system of equations deg0,ζ γ
′ = deg1,σ γ
′ = 0 is equivalent to the system
of three real equations deg0,Re ζ γ
′ = deg0,Im ζ γ
′ = deg1,σ γ
′ = 0. This system is, in turn,
equivalent to the system deg1,σ−2 Re ζ γ
′ = deg1,σ+2
√
−1 Im ζ γ
′ = 0. It remains to use Theorems 7.5
and 8.16. �
We emphasize that this result does not follow from the diffeomorphism of coarse moduli
spaces or from the equivalence of categories. Nor the diffeomorphism of coarse moduli spaces
or equivalence of categories can be derived from our result. We would also like to mention [17,
Theorem 4.2], where the equality of Voevodsky motives is proved in the case when parabolic
structures are absent and the rank and degree are coprime.
9.3 Indecomposable parabolic bundles and non-emptiness of moduli stacks
9.3.1 Indecomposable parabolic bundles
Here we recall some results of [8]. Recall that X is a smooth projective curve of genus g,
D ⊂ X(k) is a non-empty set. Let γ ∈ Γ+. We would like to know whether there exists an
indecomposable parabolic bundle of class γ. The following simple statement is noted in [7,
Introduction].
Lemma 9.5. Assume that k is algebraically closed. If g > 0, then for all γ ∈ Γ+ there is an
indecomposable parabolic bundle of class γ.
Proof. It is well-known that there is an indecomposable vector bundle on X of rank rk γ. Now
one extends it arbitrarily to a parabolic bundle of class γ. �
Next, let X = P1. Fix γ = (r, r•,•, d) ∈ Γ′ and choose a sequence of positive integers w• indexed
by D such that rx,j = 0 for j ≥ wx. Consider the star-shaped graph Gw• with vertices v∗
46 R. Fedorov, A. Soibelman and Y. Soibelman
and vx,j where x ∈ D, j is between 1 and wx − 1. The vertex v∗ is connected to all the vertices
of the form vx,1, the vertex vx,i is connected to vx,i±1 (see picture).
v∗
vx,1 vx,2 vx,wx−1
Star-shaped graph
Consider the Kac–Moody Lie algebra gw• associated to the generalized Cartan matrix defined
by this graph (see, e.g., [20, Section 1]). Let Λw• be the root lattice of gw• ; we identify it with
the free abelian group generated by the set of vertices. Then γ gives rise to an element of Λw•
given by
ργ,w• := rv∗ +
∑
x∈D
wx−1∑
j=1
(
j∑
i=1
rx,i
)
vx,j .
Now [8, p. 1334, Corollary] can be re-formulated as follows.
Proposition 9.6. In the above notation, there is a non-zero indecomposable parabolic bundle
E ∈ Bunpar
γ if and only if ργ,w• is a root of gw•.
We see that ργ,w• does not depend on d. Thus if there is an indecomposable parabolic bundle
E with cl(E) = (r, r•,•, d), then for any d′ there is an indecomposable parabolic bundle of class
(r, r•,•, d
′). Secondly, we see that the property of ργ,w• being a root does not depend on the
choice of w• as long as the components of w• are large enough. By a slight abuse of terminology
we say that γ is a root in this case.
Remark 9.7. In fact, one can consider an infinite star-shaped graphGD with degD infinite rays,
and the corresponding Kac–Moody Lie algebra gD, which is the inductive limit of gw• . Then we
have a homomorphism ρ from Γ+ to the root lattice of gD and the classes of indecomposable
parabolic bundles are exactly the ρ-preimages of roots.
9.3.2 Non-emptiness of moduli stacks
Now we can give a full answer to the question of whenHiggsσ−ss
γ (ζ), Connγ(ζ) and Conn(κ,σ)−ss
γ (ζ)
are non-empty. The first statement follows immediately from results of Crawley–Boevey.
Theorem 9.8. Assume that γ ∈ Γ+ and ζ ∈ k[D × Z>0].
(i) If g = g(X) > 0, then Connγ(ζ) is non-empty if and only if deg1,ζ γ = 0.
Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles 47
(ii) If g = 0, then Connγ(ζ) is non-empty if and only if γ can be written as
n∑
i=1
γi, where γi ∈ Γ′
are roots and deg1,ζ γi = 0.
Proof. Note that a k-stack X is non-empty if and only if X ×k k is non-empty, where k is
the algebraic closure of k. Thus, we can assume that k is algebraically closed from the very
beginning. By Lemma 8.2, Connγ(ζ) is non-empty if and only if there is a (1, ζ)-isoslopy parabolic
bundle of class γ such that deg1,ζ γ = 0. Now (i) follows from Lemma 9.5, while (ii) follows from
Proposition 9.6. �
Theorem 9.9. Assume that γ ∈ Γ+, ζ ∈ k[D × Z>0], and σ ∈ Stab.
(i) If g = g(X) > 0, then Higgsσ−ss
γ (ζ) is non-empty if and only if deg0,ζ γ = 0.
(ii) If g = 0, then Higgsσ−ss
γ (ζ) is non-empty if and only if γ can be written as
n∑
i=1
γi, where
γi ∈ Γ′ are roots, deg0,ζ γi = 0, and the (1, σ)-slope of each γi is equal to the (1, σ)-slope
of γ.
Proof. By Proposition 2.3, Higgsσ−ss
γ (ζ) is non-empty if and only if
[
Higgsσ−ss
γ (ζ)
]
6= 0. Let ζ ′
be as in Proposition 9.2. Applying Proposition 2.3 again, we see that Higgsσ−ss
γ (ζ) is non-empty
if and only if Connγ(ζ ′) is non-empty. It remains to use Theorem 9.8 and (9.2). �
Theorem 9.10. Assume that γ ∈ Γ+, ζ ∈ k[D×Z>0] and (κ, σ) ∈ Stab′. Assume that either ζ
is non-resonant, or κ = 1 and σ ∈ Stab.
(i) If g = g(X) > 0, then Conn(κ,σ)−ss
γ (ζ) is non-empty if and only if deg1,ζ γ = 0.
(ii) If g = 0, then Conn(κ,σ)−ss
γ (ζ) is non-empty if and only if γ can be written as
n∑
i=1
γi, where
γi ∈ Γ′ are roots, deg1,ζ γi = 0, and the (κ, σ)-slope of each γi is equal to the (κ, σ)-slope
of γ.
Proof. Same as of Theorem 9.9 except that one uses Proposition 9.1 instead of Proposition 9.2
and (9.1) instead of (9.2). �
Acknowledgements
We thank E. Diaconescu, J. Heinloth, O. Schiffmann, and especially A. Mellit for useful dis-
cussions and correspondence. We thank P. Boalch for a useful comment on an earlier version.
A part of this work was done while R.F. was visiting Max Planck Institute of Mathematics
in Bonn, and a part when he was visiting A. Mellit at the University of Vienna. The work
of R.F. was partially supported by NSF grant DMS–1406532. A.S. and Y.S. thank IHES for
excellent research conditions and hospitality. The work of Y.S. was partially supported by NSF
grants and Munson–Simu Faculty Award at Kansas State University. The authors would like to
thank the anonymous referees for carefully reading the paper and for useful comments.
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1 Introduction and main results
1.1 Overview
1.2 Moduli stacks
1.2.1 Parabolic bundles with connections
1.2.2 Parabolic Higgs bundles
1.2.3 Semistable parabolic bundles with connections
1.3 Motivic Donaldson–Thomas invariants
1.4 Explicit formulas
1.5 Aftermath
1.6 Other results
1.7 Other relations with previous work
1.8 Organization of the article
2 Preliminaries
2.1 Conventions
2.1.1 Partitions and nilpotent matrices
2.1.2 Stacks
2.2 Motivic functions and motivic classes
2.3 Principal bundles and special groups
3 Parabolic bundles
3.1 Definitions and notations
3.2 Monoids + and '+
3.3 Categories over Bunpar and +-graded stacks
3.4 Motivic zeta-functions and plethystic operations
3.5 Parabolic subbundles and quotient bundles
3.6 Generalized degrees and slopes
3.7 Parabolic weights and stability conditions
4 Parabolic pairs
4.1 Parabolic pairs and their generic Jordan types
4.2 Motivic classes of parabolic bundles with nilpotent endomorphisms
4.3 Factorization of graded motivic classes of stacks of nilpotent parabolic pairs
4.3.1 Jets
4.3.2 Local stacks
4.3.3 Preparation for the proof of Theorem 4.4
4.3.4 The restriction to the formal neighborhood of x
4.3.5 Proof of Proposition 4.10
4.4 Case of ¶1 and two points
4.5 Motivic modified Macdonald polynomials
4.6 Explicit formulas for the graded motivic classes of nilpotent pairs
4.7 Case of ¶1
5 Parabolic Higgs bundles with fixed eigenvalues
5.1 Stacks of Higgs bundles
5.2 Existence of Higgs fields with prescribed residues
5.3 Parabolic Higgs bundles with fixed eigenvalues
5.4 Parabolic pairs with isoslopy underlying parabolic bundles
5.5 Case of ¶1
6 Stability conditions for Higgs bundles
6.1 Harder–Narasimhan filtration
6.2 Kontsevich–Soibelman factorization formula
7 Stabilization
7.1 Stabilization of semistable Higgs bundles
7.2 Case of ¶1
8 Motivic classes of parabolic connections
8.1 Stacks of parabolic connections
8.2 Stabilization of isoslopy parabolic bundles
8.2.1 Case of ¶1
8.3 Stability conditions for bundles with connections
9 Equalities of motivic classes and non-emptiness of moduli stacks
9.1 Equalities between motivic classes of stacks
9.2 Simpson's non-abelian Hodge theory
9.3 Indecomposable parabolic bundles and non-emptiness of moduli stacks
9.3.1 Indecomposable parabolic bundles
9.3.2 Non-emptiness of moduli stacks
References
|
| id | nasplib_isofts_kiev_ua-123456789-210778 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T09:13:25Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Fedorov, Roman Soibelman, Alexander Soibelman, Yan 2025-12-17T14:37:10Z 2020 Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve. Roman Fedorov, Alexander Soibelman and Yan Soibelman. SIGMA 16 (2020), 070, 49 pages 1815-0659 2020 Mathematics Subject Classification: 14D23; 14N35; 14D20 arXiv:1910.12348 https://nasplib.isofts.kiev.ua/handle/123456789/210778 https://doi.org/10.3842/SIGMA.2020.070 Let be a smooth projective curve over a field of characteristic zero, and let be a non-empty set of rational points of . We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on ( , ) and the motivic classes of moduli stacks of semistable parabolic Higgs bundles on ( , ). As a by-product, we give a criterion for the non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem. We thank E. Diaconescu, J. Heinloth, O. Schi mann, and especially A. Mellit for useful discussions and correspondence. We thank P. Boalch for a useful comment on an earlier version. A part of this work was done while R.F. was visiting the Max Planck Institute of Mathematics in Bonn, and a part when he was visiting A. Mellit at the University of Vienna. The work of R.F. was partially supported by NSF grant DMS1406532. A.S. and Y.S. thank IHES for excellent research conditions and hospitality. The work of Y.S. was partially supported by NSF grants and the Munson-Simu Faculty Award at Kansas State University. The authors would like to thank the anonymous referees for carefully reading the paper and for their useful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve Article published earlier |
| spellingShingle | Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve Fedorov, Roman Soibelman, Alexander Soibelman, Yan |
| title | Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve |
| title_full | Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve |
| title_fullStr | Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve |
| title_full_unstemmed | Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve |
| title_short | Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve |
| title_sort | motivic donaldson-thomas invariants of parabolic higgs bundles and parabolic connections on a curve |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210778 |
| work_keys_str_mv | AT fedorovroman motivicdonaldsonthomasinvariantsofparabolichiggsbundlesandparabolicconnectionsonacurve AT soibelmanalexander motivicdonaldsonthomasinvariantsofparabolichiggsbundlesandparabolicconnectionsonacurve AT soibelmanyan motivicdonaldsonthomasinvariantsofparabolichiggsbundlesandparabolicconnectionsonacurve |