# Cohomology rings of the moduli of one-dimensional sheaves on the projective
plane

Yakov Kononov  Yale University, Department of Mathematics
<EMAIL_ADDRESS>,  Woonam Lim  Utrecht University, Department of
Mathematics<EMAIL_ADDRESS>,  Miguel Moreira  Massachusetts Institute of
Technology, Department of Mathematics<EMAIL_ADDRESS>and  Weite Pi  Yale
University, Department of Mathematics<EMAIL_ADDRESS>
###### Abstract.

We initiate a systematic study on the cohomology rings of the moduli stack
$\mathfrak{M}_{d,\chi}$ of semistable one-dimensional sheaves on the
projective plane. We introduce a set of tautological relations of geometric
origin, including Mumford-type relations, and prove that their ideal is
generated by certain primitive relations via the Virasoro operators. Using BPS
integrality and the computational efficiency of Virasoro operators, we show
that our geometric relations completely determine the cohomology rings of the
moduli stacks up to degree $5$.

As an application, we verify the refined Gopakumar–Vafa/Pandharipande–Thomas
correspondence for local $\mathbb{P}^{2}$ in degree $5$. Furthermore, we
propose a substantially strengthened version of the $P=C$ conjecture,
originally introduced by Shen and two of the authors. This can be viewed as an
analogue of the $P=W$ conjecture in a compact and Fano setting.

###### Key words and phrases:

Cohomology rings and relations, Virasoro representations, BPS integrality,
perverse filtrations, moduli spaces of sheaves.

###### Contents

  1. 0 Introduction
  2. 1 Moduli of sheaves and descendent algebras
  3. 2 Geometric relations
  4. 3 Virasoro representations
  5. 4 BPS integrality
  6. 5 Computation of the cohomology rings
  7. 6 $P=C$ and other conjectures
  8. A An identity in the quadratic descendent algebra

##  0\. Introduction

###  0.1. Overview.

The primary purpose of this paper is twofold. First, we initiate a systematic
study on the cohomology rings of the moduli stack $\mathfrak{M}_{d,\chi}$ of
semistable one-dimensional sheaves on $\mathbb{P}^{2}$, combining a variety of
ideas and tools including geometric relations, Virasoro representations, and
BPS integrality. Our methods produce explicit presentations of a number of new
cohomology rings in degrees $d\leq 5$. Second, we propose a substantially
strengthened version of the $P=C$ conjecture introduced by Shen and two of the
authors [KPS23], extending it to all cohomological degrees. This can be viewed
as an analogue of the $P=W$ conjecture in a compact, Fano setting, and has
various structural implications on both sides of the equality. We verify this
conjecture, among other predictions, on the cohomology rings we obtain.
Throughout, we work over the complex numbers $\mathbb{C}$.

Fix two integers $d$ and $\chi$ with $d\geq 1$. We consider the moduli stack
of sheaves

$\mathfrak{M}_{d,\chi}:=\\{\textrm{semistable
}{F}\in\mathrm{Coh}(\mathbb{P}^{2})\mid[\mathrm{supp}({F})]=d\cdot
H,\;\;\chi({F})=\chi\\},$

where $H$ is the class of a line, $\mathrm{supp}({F})$ denotes the Fitting
support, and the stability condition is with respect to the slope

$\mu({F})=\frac{\chi({F})}{c_{1}({F})\cdot H}\in\mathbb{Q}.$

This moduli stack is smooth and admits a good moduli space $M_{d,\chi}$
parametrizing semistable sheaves up to S-equivalence.

Our study of ${\mathfrak{M}}_{d,\chi}$ and its cohomology is motivated in part
by cohomological Hall algebras. Cohomological Hall algebras (CoHA in short)
are rich algebraic structures coming from various moduli stacks, see
Kontsevich–Soibelman [KS11]. Indeed, for a fixed slope $\mu\in{\mathbb{Q}}$
and the corresponding monoid of topological types $\Lambda_{\mu}$, the direct
sum of the cohomology groups

$\bigoplus_{(d,\chi)\in\Lambda_{\mu}}H^{*}({\mathfrak{M}}_{d,\chi},{\mathbb{Q}})$

underlies a CoHA whose structure is largely unknown. Therefore, studying
$H^{*}({\mathfrak{M}}_{d,\chi},{\mathbb{Q}})$ may provide the first step
towards understanding this CoHA. Furthermore, when $d$ and $\chi$ are coprime,
cohomology ring of the moduli stack recovers that of the moduli space via the
isomorphism

(1) $H^{*}({\mathfrak{M}}_{d,\chi},\mathbb{Q})\simeq
H^{*}(M_{d,\chi},\mathbb{Q})\otimes H^{*}(B\mathbb{G}_{m},\mathbb{Q})$

induced from the good moduli map $\pi:\mathfrak{M}_{d,\chi}\to M_{d,\chi}$,
which is a trivial $B\mathbb{G}_{m}$-gerbe in this case. Thus our study on the
cohomology of ${\mathfrak{M}}_{d,\chi}$ subsumes the coprime moduli spaces
$M_{d,\chi}$ as particularly interesting cases, which we motivate below. In
particular, we remark that our approach is inductive and understanding the
cohomology of $M_{d,\chi}$ makes use of the moduli stacks
${\mathfrak{M}}_{d^{\prime},\chi^{\prime}}$ with non-coprime
$(d^{\prime},\chi^{\prime})$ as well.

The study of the good moduli spaces $M_{d,\chi}$ dates back to the works of
Simpson [Sim94] and Le Potier [LP93a]. We refer to [PS23, Introduction] for a
brief overview of the basic properties of $M_{d,\chi}$ and various directions
to study this moduli space. From our perspective, the main motivation to
investigate $M_{d,\chi}$ comes from enumerative geometry. Briefly speaking, a
certain perverse filtration on the intersection cohomology
$\mathit{IH}^{*}(M_{d,\chi})$ defines interesting curve counting invariants
called refined BPS invariants for local ${\mathbb{P}}^{2}$, i.e. the
Calabi–Yau threefold $\mathrm{Tot}(K_{\mathbb{P}^{2}})$. These invariants are
expected to recover curve counting invariants from other enumerative theories
[PT14]. We refer to [KPS23] for a detailed account on the history and
development of this proposal, and to [KPS23, Section 1.1] for generalities
concerning the perverse filtration. When $d$ and $\chi$ are coprime, the
moduli space $M_{d,\chi}$ is a smooth projective variety and its intersection
cohomology coincides with singular cohomology. Furthermore, the perverse
filtration in this case is characterized by the cup product with a base ample
class, cf. Section 6.1, which is in turn determined by the cohomology ring
structure.

The above motivations bring us to the following problem:

######  Problem 0.1.

Study the cohomology rings of the moduli stacks ${\mathfrak{M}}_{d,\chi}$.

In this paper, we shall approach Problem 0.1 uniformly in the language of
stacks. As we will see, this is the more general setting for various
constructions, and naturally relates to the (not necessarily coprime) moduli
spaces via the BPS integrality formula.

The stack $\mathfrak{M}_{d,\chi}$ admits natural tautological classes coming
from its universal family. We will prove that these classes generate the
rational cohomology ring, cf. Theorem 1.3. These are the ring generators of
$H^{*}(\mathfrak{M}_{d,\chi},\mathbb{Q})$ we use throughout the paper.
Furthermore, we formulate in Section 2 certain geometric relations, including
notably Mumford-type relations, among these classes. The main result of this
paper, along the lines of Problem 0.1, can be stated as follows.

######  Theorem 0.2.

The geometric relations in Section 2 determine the cohomology rings of the
moduli stacks $\mathfrak{M}_{d,\chi}$ for $d\leq 4$ and the smooth (i.e.
coprime) moduli spaces $M_{5,\chi}$.

######  Remark 0.3.

(i) The main result of [LMP23], cf. Theorem 1.7, states that the cohomology
rings of $M_{d,\chi}$ with $\gcd(d,\chi)=1$ are in general $\chi$-dependent.
Thus for example we obtain two non-isomorphic cohomology rings for $M_{5,1}$
and $M_{5,2}$, respectively. (ii) According to [ES93, Mar07] and Theorem 4.4
in the case of stacks, the cycle class maps are isomorphisms for the coprime
moduli space $M_{d,\chi}$ and the stack ${\mathfrak{M}}_{d,\chi}$. Thus
Theorem 0.2, and more generally all structural results for cohomology rings in
this paper, hold for the Chow rings as well.

For $d\leq 4$ coprime to $\chi$, the cohomology rings of the stacks recover
those of the moduli spaces via the isomorphism (1); the latter are previous
known via classical or birational methods [LP93a, CM17], but our approach
recovers them independently. All other cohomology rings in Theorem 0.2, i.e.,
those of

${\mathfrak{M}}_{2,0},\;{\mathfrak{M}}_{3,0},\;{\mathfrak{M}}_{4,2},\;{\mathfrak{M}}_{4,0},\;M_{5,1},\;M_{5,2}$

are new to the best of our knowledge.111We restrict to $0\leq\chi\leq d/2$
here since these are the essential cases, cf. Section 5. We do not treat the
moduli stack $\mathfrak{M}_{5,0}$ in full, but for a purely computational
reason; see Section 5.2.7 for some partial results. For the same reason we do
not treat moduli stacks (and spaces) with $d\geq 6$, but it is reasonable to
expect that our approach would produce complete sets of relations (at least)
in low cohomological degrees, see for example [LMP23, Proposition 2.7]. To
prove Theorem 0.2, we shall introduce and utilize the ideas of Virasoro
representations and BPS integrality in Sections 3 and 4, proving independently
interesting results and leading to a systematic strategy to compute the
cohomology rings, outlined in Section 5.1. The presentations of the cohomology
rings are given in Section 5.2.222As a first indication of their complexity,
we note that $\dim H^{*}(M_{5,\chi})=1695$ for $\chi$ coprime to 5.

The last section is somewhat independent from the others, in which we
strengthen the (local) $P=C$ conjecture of [KPS23] to all cohomological
degrees, and verify a number of non-trivial predictions on the cohomology
rings. The $P=C$ conjecture relates two filtrations of highly different nature
on the cohomology of $M_{d,\chi}$ with $\gcd(d,\chi)=1$: the perverse
filtration $P_{\bullet}H^{*}(M_{d,\chi},{\mathbb{Q}})$ is defined in terms of
the topology of a fibration from $M_{d,\chi}$ to the linear system $|d\cdot
H|$ of degree $d$ planar curves, whose generic fiber is an abelian variety;
the Chern filtration $C_{\bullet}H^{*}(M_{d,\chi},{\mathbb{Q}})$, on the other
hand, is defined explicitly via the tautological classes.

######  Conjecture 0.4.

For coprime $d\geq 1$ and $\chi\in{\mathbb{Z}}$, we have
$P_{\bullet}H^{*}(M_{d,\chi},\mathbb{Q})=C_{\bullet}H^{*}(M_{d,\chi},\mathbb{Q})$.

As consequences of this conjecture, structures of either the perverse or the
Chern filtration can be transported to the other side. In particular,
Conjecture 0.4 would imply the multiplicativity of the perverse filtration and
a curious Hard Lefschetz symmetry for the Chern filtration. See Remark 6.7 for
a detailed discussion.

Since Theorem 0.2 determines the cohomology rings of the coprime moduli spaces
up to degree $5$, we are able to verify numerous predictions in this range:
the (strengthened) $P=C$ conjecture, the (refined)
Gopakumar–Vafa/Pandharipande–Thomas correspondence, and a conjecture of
Chung–Moon [CM17]. See Section 6 for the precise statements.

In what follows, we briefly discuss the main ingredients in this paper.

###  0.2. Mumford and geometric relations.

The original Mumford relations are formulated over $M_{r,d}(C)$, the moduli
space of semistable holomorphic bundles of rank $r$ and degree $d$ over a
fixed algebraic curve $C$ of genus $g\geq 2$. Mumford conjectured that such
relations form a complete set in the rank two case [AB83]. A similar but more
general construction of the relations called generalized Mumford relations is
carried out in [EK04].

Roughly speaking, the Mumford-type relations are obtained by constructing
certain vector bundles built from the universal family over the moduli spaces,
and taking the (vanishing) Chern classes of degrees beyond the rank of the
bundle. Grothendieck–Riemann–Roch theorem expresses the Chern classes in terms
of tautological classes on the moduli space, thus giving relations in the
cohomology ring. More precisely, the constructions in [AB83] and [EK04] relies
on the following vanishing results:

  1. (i)

Higher cohomology of the bundles due to positivity.

  2. (ii)

Hom (and Ext) groups between semistable bundles due to slope differences.

In the following, we reserve the name Mumford relations ($\mathsf{MR}$) for
the first type, and call the second type generalized Mumford relations
($\mathsf{GMR}$) in accordance with the original terminology. For the moduli
space $M_{r,d}(C)$, Mumford’s conjecture in the rank two case was first proved
by Kirwan [Kir92], while Earl–Kirwan [EK04] showed that the generalized
Mumford relations are complete for general rank coprime to the degree.

The study of Mumford-type relations for sheaves on (del Pezzo) surfaces was
initiated in [PS23] and played a key role in the proof of [LMP23, Theorem
1.2]; see Remark 2.13. One primary goal of the current paper is to pursue this
idea more systematically. In Section 2, we carry out general constructions of
the two Mumford-type relations for the moduli stack $\mathfrak{M}_{d,\chi}$.
In addition, there is a third type of relations, which we name base relations
($\mathsf{BR}$), coming from the fibration structure of
$\mathfrak{M}_{d,\chi}$ over the base $|d\cdot H|$. We shall refer to all of
these as geometric relations. In light of the results in [Kir92, EK04], it is
natural to ask the following question:

######  Question 0.5.

Are the geometric relations complete for general $\mathfrak{M}_{d,\chi}$?

The answer to Question 0.5 turns out to be positive for all moduli stacks (and
spaces) in Theorem 0.2, except for $M_{5,1}$. In fact, we will see that
Mumford and generalized Mumford relations are already complete for these
moduli stacks. The situation for $M_{5,1}$ is more subtle: the geometric
relations are, somewhat surprisingly, not complete in the literal sense, but
they are still sufficient to determine the entire ring structure when combined
with Poincaré duality. We refer to Section 5.2.5 for a detailed explanation.

###  0.3. Virasoro representations.

The Virasoro constraints predict a rich set of relations among tautological
invariants of moduli spaces in various settings. It is named so because the
predicted relations are described by certain representations of the half of
the Virasoro algebra
$\mathsf{Vir}_{\geq-1}:=\textup{span}\\{{\mathsf{L}}_{n}\,|\,n\geq-1\\}$,
where $\mathsf{L}_{n}$ are certain operators (cf. Section 3.1 for the
definition of these operators in our setting) satisfying the Virasoro
relations

$[\mathsf{L}_{n},\mathsf{L}_{m}]=(m-n)\,\mathsf{L}_{n+m}.$

The Witten–Kontsevich theorem [Wit91, Kon92] was the first instance of the
Virasoro constraints. This was conjecturally generalized to Gromov–Witten
theory [EHX97]. After being transported to the sheaf side via the GW/PT
correspondence [MOOP22], there have been rapid developments in the sheaf-
theoretic Virasoro constraints [Mor22, vB23, BLM22, Boj24, LM24].

For the moduli spaces of our interest, namely $M_{d,\chi}$, the Virasoro
constraint was first conjectured in [BLM22] and proven for the coprime cases
in [LM24] using quiver representations. The Virasoro constraints is a priori a
statement about top degree relations. However, it was observed in [LM24] that
the constraints imply that a representation of $\mathsf{Vir}_{\geq-1}$ on the
descendent algebra ${\mathbb{D}}_{d,\chi}$ (see Section 1.1) factors through
the surjective homomorphism333There are two representations of
$\mathsf{Vir}_{\geq-1}$ on ${\mathbb{D}}_{d,\chi}$, one from
${\mathsf{L}}_{n}$ operators and the other from ${\mathsf{R}}_{n}$ operators,
cf. Section 3.1. When we discuss the factorization property of the
$\mathsf{Vir}_{\geq-1}$-representation through
$H^{*}({\mathfrak{M}}_{d,\chi})$, we always consider the one coming from
${\mathsf{R}}_{n}$ for computational simplicity.

${\mathbb{D}}_{d,\chi}\twoheadrightarrow
H^{*}({\mathfrak{M}}_{d,\chi},{\mathbb{Q}}).$

In other words, the representation preserves the kernel of this surjection,
which we call the ideal of tautological relations. In fact, this factorization
property was proven for any ${\mathfrak{M}}_{d,\chi}$ with $d$ and $\chi$ not
necessarily coprime. Since we study the ideal of tautological relations using
geometric relations, the following question naturally arises.

######  Question 0.6.

Does the representation of $\mathsf{Vir}_{\geq-1}$ on $\mathbb{D}_{d,\chi}$
interact nicely with the ideals of geometric relations?

This question is interesting because there has been very little geometric
understanding on the Virasoro constraints. In Section 3, we shall give a
positive answer to this question in the form of the following theorem.

######  Theorem 0.7.

The $\mathsf{Vir}_{\geq-1}$-representation on ${\mathbb{D}}_{d,\chi}$
preserves the ideal of geometric relations of each type (MR, GMR, and BR).
Furthermore, in each case the entire ideal is generated by certain primitive
relations, cf. Definition 3.12, as a ${\mathbb{D}}_{d,\chi}\otimes
U(\mathsf{Vir}_{\geq-1})$-module.

Among the entire set of generators for the geometric relations, the primitive
ones form a significantly smaller subset. This brings considerable
computational efficiency when we work with the ideal of geometric relations
using a computer program.

###  0.4. BPS integrality.

One of the most subtle and important aspects in the study of tautological
rings is whether a given set of relations is complete, as inquired in Question
0.5. When the moduli space is a smooth projective variety, Poincaré duality
provides an easy criterion for the completeness of the given set of relations.
But unless we are in such a case, it is in general difficult to determine
whether the relations we have found are complete. One practical method is to
rely on the Poincaré series of the moduli stacks and spaces in question.

In this regard, the BPS integrality formula of cohomological Hall algebras
(CoHA) for hereditary categories by Mozgovoy–Reineke [MR15] is very helpful.
For a fixed slope $\mu\in{\mathbb{Q}}$, the universal extension diagram
defines a CoHA structure on the direct sum

$\bigoplus_{(d,\chi)\in\Lambda_{\mu}}H^{*}({\mathfrak{M}}_{d,\chi},{\mathbb{Q}}).$

Roughly speaking, the BPS integrality of [MR15] relates cohomology groups of
the moduli stacks ${\mathfrak{M}}_{d,\chi}$ and intersection cohomology groups
of the moduli spaces $M_{d,\chi}$ in the localized $K$-group of mixed Hodge
modules. This in particular implies an equality between their virtual Hodge
polynomials.

In order to derive an equality between the actual Poincaré series of the
moduli stacks and the intersection Poincaré series of the moduli spaces, we
need the following properties (cf. Theorem 4.4) of the cohomology of
${\mathfrak{M}}_{d,\chi}$. This is proven in Section 4 after we introduce BPS
integrality.

######  Theorem 0.8.

The cohomology ring $H^{*}({\mathfrak{M}}_{d,\chi},{\mathbb{Q}})$ is
tautologically generated and pure. Furthermore, the cycle class map from Chow
to cohomology is an isomorphism.

The above theorem in particular shows that
$H^{*}({\mathfrak{M}}_{d,\chi},{\mathbb{Q}})$ is of $(p,p)$-type. The
corresponding property for $\mathit{\mathit{IH}}^{*}(M_{d,\chi},{\mathbb{Q}})$
was proven by [Bou22]. Therefore, the BPS integrality formula implies the
following equality between the shifted (intersection) Poincaré series

(2)
$\sum_{(d,\chi)\in\Lambda_{\mu}}\overline{E}({\mathfrak{M}}_{d,\chi},q)\cdot
e^{(d,\chi)}=\mathrm{PE}\left(\frac{-q^{1/2}}{1-q}\cdot\sum_{(d,\chi)\in\Lambda_{\mu}\backslash\\{(0,0)\\}}\overline{\mathit{IE}}(M_{d,\chi},q)\cdot
e^{(d,\chi)}\right).$

where $\mathrm{PE}(-)$ denotes the plethystic exponential. Note that on the
right hand side we have moduli spaces $M_{d,\chi}$ that are not necessarily
smooth, whose (intersection) cohomology is a priori more mysterious. In this
regard, we recall the following $\chi$-independence theorem of Maulik–Shen:

######  Theorem 0.9 ([MS23, Theorem 0.1]).

For any $\chi,\chi^{\prime}\in\mathbb{Z}$ not necessarily coprime to $d$,
there is a (non-canonical) isomorphism of graded vector spaces

$\mathit{IH}^{*}(M_{d,\chi})\simeq\mathit{IH}^{*}(M_{d,\chi^{\prime}}).$

Thanks to this theorem, we can use Equation (2) to effectively compute the
Betti numbers of ${\mathfrak{M}}_{d,\chi}$ in terms of the Betti numbers of
the smooth moduli spaces $M_{d^{\prime},1}$ where
$(d,\chi)=k(d^{\prime},\chi^{\prime})$ for some $k\geq 1$. The latter can be
computed directly (at least for $d\leq 5$) using geometric relations and the
Poincaré duality criterion, see Section 5.2. Knowing the Betti numbers then
allows us to check whether our geometric relations are complete for the moduli
stacks.

###  0.5. Perverse = Chern.

We briefly recall the history and background of the ‘Perverse $=$ Chern’
phenomenon, manifested by the $P=C$ conjecture we shall discuss in Section
6.1. Very roughly, this phenomenon states that for certain abelian fibrations,
the location of a tautological class in the associated perverse filtration is
determined by its Chern grading from a universal family.

The idea traces back to Beauville and Deninger–Murre’s work [Bea86, DM91] on
abelian schemes $\pi:A\to B$. The authors used the Fourier transform
$\mathfrak{F}$ induced by a normalized Poincaré bundle $\mathcal{L}$ over
$A^{\vee}\times_{B}A$ to obtain a motivic decomposition, called Beauville
decomposition, on the cohomology of the abelian scheme that splits the
(perverse) Leray filtration. The Fourier transform is a sum with different
Chern gradings

$\mathfrak{F}=\sum_{k}\mathfrak{F}_{k}:H^{*}(A^{\vee},\mathbb{Q})\to
H^{*}(A,\mathbb{Q}),$

where $\mathfrak{F}_{k}$ is induced by the $k$-th Chern character
$\mathrm{ch}_{k}(\mathcal{L})$. It is then shown in [Bea86, DM91] that the
images of $\mathfrak{F}_{k}$ land precisely in the $k$-th graded piece in the
splitting of the Leray filtration, i.e. their ‘perversity’ equals the Chern
grading from the Poincaré line bundle.

The Beauville decomposition relies crucially on the multiplication by $N$
operator on abelian schemes, but similar $P=C$ phenomena has been found for
more general abelian fibrations with possibly singular fibers, for instance
Lagrangian fibrations associated to compact hyperkähler manifolds [SY22] and
compactified Jacobians of integral planar curves [OY17, MY14]. We refer to
[MSY23, Introduction] for a detailed discussion. Notably, the recently proved
$P=W$ conjecture of de Cataldo–Hausel–Migliorini [dCHM12, MS22, HMMS22, MSY23]
is equivalent to a $P=C$ statement for the Hitchin system associated to the
Dolbeault moduli spaces [dCMS22].

The $P=C$ phenomenon we investigate in this paper arises from the Hilbert–Chow
morphism, cf. Section 1.2 (6), of the moduli spaces $M_{d,\chi}$ with
$\gcd(d,\chi)=1$. Indeed, this is a weak abelian fibration in the sense of
[MS23], and induces a perverse filtration on the total cohomology
$H^{*}(M_{d,\chi},\mathbb{Q})$. Using the (normalized) tautological generators
$c_{k}(j)$ of Proposition 1.9, Shen and two of the authors defined in [KPS23]
a natural Chern filtration

$C_{0}H^{\leq 2d-4}(M_{d,\chi})\subset C_{1}H^{\leq
2d-4}(M_{d,\chi})\subset\cdots\subset H^{\leq 2d-4}(M_{d,\chi},\mathbb{Q}).$

Furthermore, the following local444In the sense that the cohomological degree
is bounded by $2d-4$. $P=C$ conjecture [KPS23, Conjecture 0.3] was proposed:

(3) $P_{\bullet}H^{\leq 2d-4}(M_{d,\chi})=C_{\bullet}H^{\leq
2d-4}(M_{d,\chi}).$

One consequence of this is an asymptotic product formula [KPS23, Conjecture
0.1] for the refined BPS invariants for local ${\mathbb{P}}^{2}$. The equality
(3) is verified in [KPS23] for degrees $\leq 4$, and [MSY23] established one
side of the inclusion $P\supset C$. In Section 6.1, we shall explain how to
upgrade this local version into a global one, using a modified (and perhaps
more natural) definition of the Chern filtration involving all tautological
classes in addition to the generators. This removes the linear bound $2d-4$ on
cohomological degrees.

Finally, we remark that by the spectral correspondence [BNR89], Higgs bundles
on a curve $C$ are in correspondence with one-dimensional sheaves on the
surface $\mathrm{Tot}(K_{C})$. Thus the $P=C$ conjecture we propose can be
viewed as an analogue of the $P=W$ conjecture in a compact and Fano setting.
Indeed, [dCHM12, Theorem 4.2.2] compares the perverse filtration of
meromorphic Hitchin system in rank $2$ with another filtration defined by
tautological classes analogous to the Chern filtration.

###  0.6. Further directions.

We outline a few directions of future research.

  1. (i)

The relations in $H^{*}(M_{5,1},\mathbb{Q})$ that cannot be obtained as
geometric relations, cf. Section 5.2.5, seem rather mysterious to us. It is
natural to ask if one can obtain these relations by other geometric means
involving new ideas. If this can be achieved, it is then reasonable to enlarge
the set of geometric relations and study Problem 0.1 in general, with the aim
of proving a result similar to [EK04].

  2. (ii)

The $P=C$ conjecture gives various predictions on the perverse and the Chern
filtrations on $M_{d,\chi}$, see Remark 6.7. Confirming these predictions
independently would in turn give more evidence to the conjecture. In
particular, Remark 6.7 (ii) and (iv) are closely related to the theme of
finding relations in this paper. We will prove in Section 2.3 a subset of the
vanishing Chern monomials predicted by Remark 6.7 (iv), and we expect similar
methods to extend and establish more results along this line.

  3. (iii)

The conjectures we verify in Section 6 mostly concern cohomology of the moduli
spaces. One can ask what extra structures the cohomology of the stacks
$\mathfrak{M}_{d,\chi}$ admit. Indeed, with a careful formulation of the
perverse filtration on the stacks as in [Dav23, Section 1.3], we have obtained
strong numerical evidence towards a stacky $P=C$ conjecture. More precisely,
it is verified that the graded dimensions of the perverse filtration and the
Chern filtration match perfectly in all cohomological degrees for the moduli
stacks we compute, cf. Theorem 6.9. A more systematic study of this conjecture
would likely require a better understanding of the CoHA structure and the
interaction of BPS integrality with the Chern filtration. We plan to explore
this direction in the future.

We conclude with a remark on the generality of this work. We have focused
exclusively on moduli of sheaves on the surface $\mathbb{P}^{2}$, but the key
ingredients and constructions used in this paper, for example [MR15, PS23,
LM24], are available to other smooth del Pezzo surfaces. We expect that our
approach can be adapted to study the cohomology of moduli of sheaves in this
more general setting.

###  0.7. Notations and conventions

All cohomology rings $H^{*}(-)$ take $\mathbb{Q}$-coefficients if unspecfied,
and will be considered only for smooth spaces. For possibly singular spaces,
we use the intersection cohomology $\mathit{IH}^{*}(-)$. The shifted
(intersection) virtual Poincaré series, denoted by $\overline{E}(-,q)$ or
$\overline{\mathit{IE}}(-,q)$, is defined as the non-shifted one multiplied by
$(-q)^{-\dim(-)/2}$.

Here are some notations we use throughout the paper:

[label = , column-sep=-17em, item-indent=1.5em](2)
$\alpha,\>\\!\alpha^{\prime}$

Topological type of one-dimensional sheaves

$\mathbb{D},\>\\!\mathbb{D}_{\mathrm{wt}_{0}}$

Descendent algebra and the weight zero subalgebra

$\mathbb{D}_{\alpha},\>\\!\mathbb{D}_{\alpha,\mathrm{wt}_{0}}$

Descendent algebra for topological type $\alpha$

$\mathrm{ch}_{i}(\gamma)$

Formal descendent symbols

$c_{k}(j)$

Normalized tautological classes555We refer to Section 1.2.3 for the precise
relations between $\mathrm{ch}_{i}(\gamma)$ and $c_{k}(j)$.

${\mathfrak{I}}_{d,\chi},\>\\!I_{d,\chi}$

Ideal of tautological relations for ${\mathfrak{M}}_{d,\chi}$ and $M_{d,\chi}$

${\mathfrak{I}}^{\bullet}_{d,\chi},\>\\!I^{\bullet}_{d,\chi}$

Ideal of geometric relations, where $\bullet\in\\{\mathsf{MR,GMR,BR,Geom}\\}$

$\mathsf{L}_{n},\>\\!\mathsf{R}_{n}$

Virasoro operators acting on $\mathbb{D}$

$\mathsf{L}_{n}^{\delta},\>\\!\mathsf{R}_{n}^{\delta}$

Normalized Virasoro operators acting on $\mathbb{D}_{\mathrm{wt_{0}}}$

$\Lambda,\>\\!\Lambda_{\mu}$

Monoid of topological types (and of fixed slope $\mu$)

$P_{\bullet},\>\\!C_{\bullet}$

Perverse and Chern filtrations on cohomology

###  0.8. Acknowledgements.

We would like to thank Ben Davison, Andres Fernandez Herrero, Davesh Maulik,
Sebastian Schlegel Mejia, Rahul Pandharipande, Junliang Shen, and Ruijie Yang
for helpful conversations and comments. WL is supported by SNF-200020-182181
and ERC Consolidator Grant FourSurf 101087365. MM was supported during part of
the project by ERC-2017-AdG-786580-MACI. The project received funding from the
European Research Council (ERC) under the European Union Horizon 2020 research
and innovation programme (grant agreement 786580).

##  1\. Moduli of sheaves and descendent algebras

###  1.1. Descendent algebra

Our goal in this paper is to study the cohomology rings
$H^{*}({\mathfrak{M}}_{d,\chi})$ for general $(d,\chi)$ and
$H^{*}(M_{d,\chi})$ for $\gcd(d,\chi)=1$ in terms of generators and relations.
A natural set of generators can be constructed using the Chern classes of
universal sheaves. It is useful to introduce the free algebra generated by the
formal descendent symbols. We follow the notation and presentation in [BLM22],
and later we establish the connection to the normalized generators $c_{k}(j)$
used in [PS23, KPS23, LMP23].

######  Definition 1.1 (Descendent algebra).

Let ${\mathbb{D}}$ be the free commutative and unital ${\mathbb{Q}}$-algebra
generated by the formal symbols

${\mathrm{ch}}_{i}(\gamma)\,\textup{ for }\,i\geq 0,\;\gamma\in
H^{*}(\mathbb{P}^{2},\mathbb{Q})$

modulo the linearity relations

${\mathrm{ch}}_{i}(\lambda_{1}\gamma_{1}+\lambda_{2}\gamma_{2})=\lambda_{1}{\mathrm{ch}}_{i}(\gamma_{1})+\lambda_{2}{\mathrm{ch}}_{i}(\gamma_{2})\textup{\,
for }\gamma_{1},\gamma_{2}\in
H^{*}(\mathbb{P}^{2}),\,\lambda_{1},\lambda_{2}\in{\mathbb{Q}}\,.$

Given $\alpha=(d,\chi)$, we let ${\mathbb{D}}_{\alpha}$ be the quotient of
${\mathbb{D}}$ by the relations666Here, ${\mathrm{ch}}(\alpha)$ denotes the
Chern character of any one-dimensional sheaf of type $\alpha$.

${\mathrm{ch}}_{0}(\gamma)=\int_{\mathbb{P}^{2}}{\mathrm{ch}}(\alpha)\cdot\gamma,\quad\gamma\in
H^{*}(\mathbb{P}^{2}).$

We call ${\mathbb{D}}$ the descendent algebra. Using the universal sheaf
${\mathcal{F}}$ on ${\mathfrak{M}}_{d,\chi}\times\mathbb{P}^{2}$, we can
realize descendents in the cohomology on moduli stacks. Explicitly, denoting
$p$ and $q$ the respective projections to ${\mathfrak{M}}_{d,\chi}$ and to
$\mathbb{P}^{2}$, we have a map of algebras

$\xi\colon{\mathbb{D}}\to H^{*}({\mathfrak{M}}_{d,\chi})$

given by

${\mathrm{ch}}_{i}(\gamma)\mapsto\Big{[}p_{\ast}\big{(}{\mathrm{ch}}({\mathcal{F}})\cdot
q^{\ast}\gamma)\Big{]}_{2i}\in H^{2i}({\mathfrak{M}}_{d,\chi})\,$

where the notation $[-]_{2i}$ means that we extract the degree $2i$ part of a
non-homogeneous cohomology class. In particular, we have

${\mathrm{ch}}_{i}(H^{j})\mapsto
p_{\ast}\big{(}{\mathrm{ch}}_{i+2-j}({\mathcal{F}})\cdot
q^{\ast}H^{j}\big{)}\in H^{2i}({\mathfrak{M}}_{d,\chi})\,.$

Note that for a given $\alpha=(d,\chi)$, the map $\xi$ factors through a map
${\mathbb{D}}_{\alpha}\to H^{*}({\mathfrak{M}}_{d,\chi})$, which we still
denote by $\xi$.

For the good moduli space $M_{d,\chi}$ with $\gcd(d,\chi)=1$, we can only
define a map

$\xi_{\mathbb{F}}\colon{\mathbb{D}}\to H^{*}(M_{d,\chi})$

after we fix the choice of a universal sheaf ${\mathbb{F}}$, which exists by
[HL10, Corollary 4.6.7]. There is, however, a subalgebra
${\mathbb{D}}_{\mathrm{wt}_{0}}\subset{\mathbb{D}}$ on which a canonical map
is defined.

######  Definition 1.2.

Let ${\mathsf{R}}_{-1}:{\mathbb{D}}\to{\mathbb{D}}$ be a derivation defined on
generators by

${\mathsf{R}}_{-1}({\mathrm{ch}}_{i}(\gamma))={\mathrm{ch}}_{i-1}(\gamma)\,,$

where we set ${\mathrm{ch}}_{-1}(\gamma)=0$. We define

${\mathbb{D}}_{\mathrm{wt}_{0}}=\ker{\mathsf{R}}_{-1}\subset{\mathbb{D}}.$

Similarly, we can define
${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\subset{\mathbb{D}}_{\alpha}$.

As explained in [BLM22, Section 2.4], the restriction of $\xi_{{\mathbb{F}}}$
to ${\mathbb{D}}_{\mathrm{wt}_{0}}$ does not depend on the choice of
${\mathbb{F}}$, so there is a canonical map

${\mathbb{D}}_{\mathrm{wt}_{0}}\to H^{*}(M_{d,\chi})\,.$

Descendents indeed generate the cohomology of moduli spaces and moduli stacks.

######  Theorem 1.3.

Let $d,\chi\in{\mathbb{Z}}$ with $d>0$.

  1. (i)

For coprime $(d,\chi)$, the map ${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\to
H^{*}(M_{d,\chi})$ is surjective.

  2. (ii)

For general $(d,\chi)$, the map ${\mathbb{D}}_{\alpha}\to
H^{*}({\mathfrak{M}}_{d,\chi})$ is surjective.

The statement (i) for the moduli spaces follows from [PS23, Theorem 0.2 (a)]
together with the description of the generators $c_{k}(j)$ in terms of
${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$ that we will explain in Section 1.2.
The statement (ii) for the stacks is new and is the content of Theorem 4.4.

######  Definition 1.4.

Let $I_{d,\chi}$ and ${\mathfrak{I}}_{d,\chi}$ be the kernels of the maps

${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\to H^{*}(M_{d,\chi})\textup{ \,and\,
}{\mathbb{D}}_{\alpha}\to H^{*}({\mathfrak{M}}_{d,\chi})\,,$

respectively.

Studying these ideals is the main goal of this paper. We start with a basic
property of the ideals on the stack by using the geometric interpretation of
the operator ${\mathsf{R}}_{-1}$.

######  Proposition 1.5.

The operator ${\mathsf{R}}_{-1}$ preserves the ideal
${\mathfrak{I}}_{d,\chi}$. Thus it descends to
$H^{*}({\mathfrak{M}}_{d,\chi})$, i.e. there exists a morphism completing the
diagram

${{\mathbb{D}}_{\alpha}}$${{\mathbb{D}}_{\alpha}}$${H^{*}({\mathfrak{M}}_{d,\chi})}$${H^{*}({\mathfrak{M}}_{d,\chi}).}$$\scriptstyle{{\mathsf{R}}_{-1}}$$\scriptstyle{\xi}$$\scriptstyle{\xi}$

###### Proof.

We show this by constructing the operator on the cohomology of the stack that
makes the diagram commute. There is a $B{\mathbb{G}}_{m}$-action

$\mathrm{act}\colon{\mathfrak{M}}_{d,\chi}\times
B{\mathbb{G}}_{m}\to{\mathfrak{M}}_{d,\chi}$

induced by the ${\mathbb{G}}_{m}$-automorphisms of sheaves. Then we define the
map

$H^{*}({\mathfrak{M}}_{d,\chi})\xlongrightarrow{\mathrm{act}^{\ast}}H^{*}({\mathfrak{M}}_{d,\chi}\times
B{\mathbb{G}}_{m})\cong
H^{*}({\mathfrak{M}}_{d,\chi})[\zeta]\xlongrightarrow{[\zeta^{1}]}H^{*}({\mathfrak{M}}_{d,\chi})$

where the last map takes the $\zeta^{1}$ coefficient of a polynomial in
$\zeta$. The argument in [BLM22, Lemma 4.9] shows that this makes the diagram
in Proposition 1.5 commute. ∎

###  1.2. Normalized universal sheaves and generators.

Although the algebra of weight zero descendents is a very useful object from a
conceptual point of view, it is not so easy to work with in concrete
computations, for instance to describe explicitly the ideal $I_{d,\chi}$ for
small $d$.

Instead, we can normalize the universal sheaf on
$M_{d,\chi}\times\mathbb{P}^{2}$, which is the approach taken in [PS23].
There, the authors take an arbitrary universal sheaf ${\mathbb{F}}$ and modify
it by

${\mathbb{F}}^{\textup{norm}}={\mathbb{F}}\otimes e^{A_{\mathbb{F}}}$

where $A_{\mathbb{F}}\in H^{2}(M_{d,\chi}\times\mathbb{P}^{2},{\mathbb{Q}})$
is chosen in a way such that ${\mathbb{F}}^{\textup{norm}}$ becomes
independent of ${\mathbb{F}}$. More precisely, we have

######  Definition 1.6.

Given a universal sheaf ${\mathbb{F}}$, let $A_{\mathbb{F}}\in
H^{2}(M_{d,\chi}\times\mathbb{P}^{2},{\mathbb{Q}})$ be the class in [KPS23,
Lemma 2.5] and define the normalized classes

(4)
$c_{k}(j):=p_{\ast}\big{(}{\mathrm{ch}}_{k+1}({\mathbb{F}}^{\textup{norm}})\cdot
q^{\ast}H^{j}\big{)}\in H^{2k+2j-2}(M_{d,\chi}),\quad j\in\\{0,1,2\\}$

where ${\mathrm{ch}}_{k+1}({\mathbb{F}}^{\textup{norm}})$ is the degree $2k+2$
part of ${\mathrm{ch}}({\mathbb{F}})\cdot e^{A_{\mathbb{F}}}$.

The class $A_{\mathbb{F}}$ is the unique class such that the following holds
[PS23, Proposition 1.3]:

(5) $c_{1}(0)=0\in H^{0}(M_{d,\chi})\textup{ \,and\, }c_{1}(1)=0\in
H^{2}(M_{d,\chi})\,.$

Moreover, the normalized universal sheaf ${\mathbb{F}}^{\mathrm{norm}}$, and
hence the classes $c_{k}(j)$, does not depend on the choice of ${\mathbb{F}}$.

These normalized tautological classes have nice properties established in
[PS23, KPS23, Yua24]. Before stating them, we introduce two more structures on
the moduli space $M_{d,\chi}$. First, it admits a proper and flat Hilbert–Chow
map

(6) $h:M_{d,\chi}\to|d\cdot
H|=\mathbb{P}^{b},\quad{\mathcal{F}}\mapsto\mathrm{supp}({\mathcal{F}})$

sending a sheaf to its Fitting support, where

$b=\dim H^{0}(\mathbb{P}^{2},\mathcal{O}(d))-1=\frac{1}{2}d(d+3).$

This is a weakly abelian fibration in the sense of [MS23], and a general fiber
of $h$ is a Picard variety whose dimension is $g=\frac{1}{2}(d-1)(d-2)$.
Second, the moduli space $M_{d,\chi}$ (and similarly the stacks
${\mathfrak{M}}_{d,\chi}$) admits two types of symmetry:

  1. (a)

The first type is given by the isomorphism

$\psi_{1}:M_{d,\chi}\xrightarrow{\sim}M_{d,\chi+d},\;\;\mathcal{F}\mapsto\mathcal{F}\otimes\mathcal{O}_{\mathbb{P}^{2}}(1).$

  2. (b)

The second type [Mai10, Theorem 13] is given by the duality isomorphism

$\psi_{2}:M_{d,\chi}\xrightarrow{\sim}M_{d,-\chi},\;\;\mathcal{F}\mapsto{\mathcal{E}}\kern-1.5pt\mathit{xt}^{1}({\mathcal{F}},\omega_{{\mathbb{P}}^{2}}).$

The following $\chi$-dependence theorem is proved in [LMP23]. It provides
context to our main results and contrast with the $\chi$-independence of the
(intersection) Betti numbers of $M_{d,\chi}$, cf. Theorem 0.9.

######  Theorem 1.7.

For $d\geq 1$ and $\chi,\chi^{\prime}$ coprime to $d$, there is an isomorphism
of graded $\mathbb{Q}$-algebras

$H^{*}(M_{d,\chi})\simeq H^{*}(M_{d,\chi^{\prime}})$

if and only if $\chi$ and $\chi^{\prime}$ are related by the two symmetries
above.

Now we return to the normalized tautological classes $c_{k}(j)$.

######  Proposition 1.8 ([PS23, Section 1.2]).

With the above notations,

  1. (i)

The class $c_{0}(2)$ is the pull-back of the hyperplane class via $h$, and
$c_{2}(0)$ is relatively ample. These two classes span
$H^{2}(M_{d,\chi},\mathbb{Q})$ for $d\geq 3$.

  2. (ii)

We have

$\psi_{1}^{*}c_{k}(j)=c_{k}(j),\quad\psi_{2}^{*}c_{k}(j)=(-1)^{k}c_{k}(j).$

######  Theorem 1.9 ([PS23, Yua24]).

Assume $d\geq 3$. We have:

  1. (i)

$H^{*}(M_{d,\chi})$ is generated as a ${\mathbb{Q}}$-algebra by the $3d-7$
classes777We call these $3d-7$ classes normalized tautological generators. of
degrees $\leq 2d-4$:

$c_{0}(2),\,c_{2}(0)\in
H^{2}(M_{d,\chi}),\;\;c_{k}(0),\,c_{k-1}(1),\,c_{k-2}(2)\in
H^{2k-2}(M_{d,\chi}),\;\;3\leq k\leq d-1.$

  2. (ii)

There are no relations among these $3d-7$ classes in degrees $\leq 2d-2$, and
exactly three linearly independent relations in degree $2d$ if $d\geq 5$.

######  Remark 1.10.

One reason we introduce the normalized classes $c_{k}(j)$ is that the
vanishing of $c_{1}(0)$ and $c_{1}(1)$ simplifies computations and we will use
them to present our cohomology rings. A more important motivation is that this
choice of normalization gives the correct formulation of the local $P=C$
conjecture (3); see [KPS23, Proposition 1.2]. In fact, the choice of
normalization in the definition of $c_{k}(j)$ has been influenced by the $P=W$
literature, where similar normalized classes on the Dolbeault moduli spaces
are denoted by $c_{k}(\gamma)$, see for example [MS22]. A more canonical and
normalization-free definition of the Chern filtration will be given in Section
6.1.2.

We now proceed to understand the generators $c_{k}(j)$ in terms of the
canonical map ${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\to H^{*}(M_{d,\chi})$.
The class $A_{\mathbb{F}}$ can be decomposed as a sum of classes in
$H^{2}(M_{d,\chi})$ and $H^{2}(\mathbb{P}^{2})$; we study the effect of
twisting by a line bundle on $M_{d,\chi}$ and on $\mathbb{P}^{2}$ separately,
following ideas in [BLM22, Sections 2.5 and 2.7].

####  1.2.1. Twist by line bundles on $M_{d,\chi}$.

Let $\alpha=(d,\chi)\in K({\mathbb{P}}^{2})$ and let $\delta=H/d\in
H^{*}(\mathbb{P}^{2})$. This choice has the property that

${\mathrm{ch}}_{0}(\delta)=\int_{{\mathbb{P}}^{2}}{\mathrm{ch}}(\alpha)\cdot\delta=1\in{\mathbb{D}}_{\alpha}\,.$

We make the following definition:

######  Definition 1.11.

For any universal sheaf ${\mathbb{F}}$, we let

${\mathbb{F}}_{\delta}={\mathbb{F}}\otimes
e^{-\xi_{\mathbb{F}}({\mathrm{ch}}_{1}(\delta))}\,.$

We call ${\mathbb{F}}_{\delta}$ the $\delta$-normalized universal sheaf, which
is characterized by the property that

$\xi_{{\mathbb{F}}_{\delta}}({\mathrm{ch}}_{1}(\delta))=0\,.$

Note that ${\mathbb{F}}_{\delta}$ does not depend on the initial choice of
${\mathbb{F}}$. Indeed, if ${\mathbb{F}}^{\prime}={\mathbb{F}}\otimes
p^{\ast}L$ then

${\mathbb{F}}^{\prime}\otimes
e^{-\xi_{{\mathbb{F}}^{\prime}}({\mathrm{ch}}_{1}(\delta))}={\mathbb{F}}^{\prime}\otimes
e^{-\xi_{{\mathbb{F}}}({\mathrm{ch}}_{1}(\delta))-c_{1}(L)\xi_{{\mathbb{F}}}({\mathrm{ch}}_{0}(\delta))}={\mathbb{F}}\otimes
e^{-\xi_{{\mathbb{F}}}({\mathrm{ch}}_{1}(\delta))}\,.$

The effect of normalizing on descendents is controlled by the operator

$\eta=\eta_{\delta}\colon\,{\mathbb{D}}_{\alpha}\to{\mathbb{D}}_{\alpha}$

given by

$\eta=\sum_{j\geq
0}\frac{(-1)^{j}}{j!}{\mathrm{ch}}_{1}(\delta)^{j}{\mathsf{R}}_{-1}^{j}\,.$

######  Proposition 1.12.

Let $\alpha\in K(\mathbb{P}^{2})$ and $\delta\in H^{*}(\mathbb{P}^{2})$ be
such that $\int_{\mathbb{P}^{2}}{\mathrm{ch}}(\alpha)\cdot\delta=1$, and let
$\eta=\eta_{\delta}$ be the operator above. We have the following:

  1. (i)

${\mathsf{R}}_{-1}\circ\eta=0$, so

$\eta\colon{\mathbb{D}}_{\alpha}\to{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\,.$

  2. (ii)

$\eta$ is an algebra homomorphism.

  3. (iii)

$\eta$ is the identity when restricted to
${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\subset{\mathbb{D}}_{\alpha}$, and
$\eta({\mathrm{ch}}_{1}(\delta))=0$.

  4. (iv)

The composition

${\mathbb{D}}_{\alpha}\xlongrightarrow{\eta}{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\longrightarrow
H^{*}(M)$

is precisely $\xi_{{\mathbb{F}}_{\delta}}$.

###### Proof.

  1. (i)

We have

$[{\mathsf{R}}_{-1},{\mathrm{ch}}_{1}(\delta)^{j}]=j[{\mathsf{R}}_{-1},{\mathrm{ch}}_{1}(\delta)]{\mathrm{ch}}_{1}(\delta)^{j-1}=j{\mathrm{ch}}_{0}(\delta){\mathrm{ch}}_{1}(\delta)^{j-1}=j{\mathrm{ch}}_{1}(\delta)^{j-1}$

where $[-,-]$ denotes commutator of operators and we write
${\mathrm{ch}}_{1}(\delta)^{j}$ for the operator of multiplication by
${\mathrm{ch}}_{1}(\delta)^{j}\in{\mathbb{D}}_{\alpha}$. After applying the
commutator above, the sum ${\mathsf{R}}_{-1}\circ\eta$ telescopes and we
obtain zero.

  2. (ii)

This is a formal consequence of the fact that ${\mathsf{R}}_{-1}$ is a
derivation. Indeed, by the general Leibniz rule,

$\displaystyle\eta(D_{1}D_{2})=\sum_{j\geq
0}\frac{(-1)^{j}}{j!}{\mathrm{ch}}_{1}(\delta)^{j}\sum_{s+t=j}\binom{j}{s}{\mathsf{R}}_{-1}^{s}(D_{1}){\mathsf{R}}_{-1}^{t}(D_{2})=\eta(D_{1})\eta(D_{2})\,.$

  3. (iii)

Both claims are immediate.

  4. (iv)

Apply [BLM22, Lemma 2.8], specializing to
$\zeta=-\xi_{{\mathbb{F}}}({\mathrm{ch}}_{1}(\delta))$.∎

The idea of $\delta$-normalization and the operator $\eta$ allow for a more
concrete description of ${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$. By
(i)-(iii) in Proposition 1.12, $\eta$ induces an algebra homomorphism

$\eta\colon{\mathbb{D}}_{\alpha}/\langle{\mathrm{ch}}_{1}(\delta)\rangle\to{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\,.$

We show that this map is an isomorphism in the next proposition. For this, we
define algebra homomorphisms

$\phi\colon{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}[u]\to{\mathbb{D}}_{\alpha}\textup{
and
}\varphi\colon{\mathbb{D}}_{\alpha}\to{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}[u]$

as follows: $\phi$ restricted to ${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$ is
the inclusion
${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\hookrightarrow{\mathbb{D}}_{\alpha}$
and $\phi(u)={\mathrm{ch}}_{1}(\delta)$, while

$\varphi=\sum_{i\geq 0}\frac{u^{i}}{i!}\eta\circ{\mathsf{R}}_{-1}^{i}\,.$

The same proof that $\eta$ is an algebra homomorphism shows that $\varphi$ is
as well.

######  Proposition 1.13.

The homomorphisms $\phi$ and $\varphi$ are inverse isomorphisms. In
particular,

$\eta\colon{\mathbb{D}}_{\alpha}/\langle{\mathrm{ch}}_{1}(\delta)\rangle\to{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\,$

is an isomorphism.

###### Proof.

We need to show that both $\varphi\circ\phi$ and $\phi\circ\varphi$ are the
identity. By Proposition 1.12 (iii), $\varphi\circ\phi$ is the identity when
restricted to ${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$. Moreover,

$\varphi(\phi(u))=\varphi({\mathrm{ch}}_{1}(\delta))=\eta({\mathrm{ch}}_{1}(\delta))+u\cdot\eta({\mathrm{ch}}_{0}(\delta))=u\,.$

For $\phi\circ\varphi$ we have

$\phi\circ\varphi=\sum_{i\geq 0}\sum_{j\geq
0}\frac{{\mathrm{ch}}_{1}(\delta)^{i}}{i!}\frac{(-1)^{j}{\mathrm{ch}}_{1}(\delta)^{j}}{j!}{\mathsf{R}}_{-1}^{i+j}=\textup{id}$

by the binomial formula. The statement for $\eta$ follows from the commutative
diagram

${{\mathbb{D}}_{\alpha}}$${{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}[u]}$${{\mathbb{D}}_{\alpha}/\langle{\mathrm{ch}}_{1}(\delta)\rangle}$${{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\,}$$\scriptstyle{\varphi}$$\scriptstyle{\eta}$

where the arrow on the right sends $u$ to zero. ∎

####  1.2.2. Twist by line bundles on $\mathbb{P}^{2}$.

Given $\rho\in H^{2}(\mathbb{P}^{2})$, let
${\mathsf{F}}_{\rho}\colon{\mathbb{D}}\to{\mathbb{D}}$ be the algebra
isomorphism given by

${\mathsf{F}}_{\rho}\big{(}{\mathrm{ch}}_{i}(\gamma)\big{)}={\mathrm{ch}}_{i}(e^{\rho}\gamma)\,.$

Given any universal sheaf ${\mathbb{F}}$ on $M_{d,\chi}\times\mathbb{P}^{2}$,
we have

$\xi_{{\mathbb{F}}\otimes
e^{\rho}}=\xi_{\mathbb{F}}\circ{\mathsf{F}}_{\rho}\,.$

Note that ${\mathsf{F}}_{\rho}$ commutes with ${\mathsf{R}}_{-1}$, so it is
also well-defined as a map
${\mathbb{D}}_{\mathrm{wt}_{0}}\to{\mathbb{D}}_{\mathrm{wt}_{0}}$. It can also
be easily checked that ${\mathsf{F}}_{\rho}$ induces

${\mathsf{F}}_{\rho}\colon{\mathbb{D}}_{e^{\rho}\cdot\alpha}\to{\mathbb{D}}_{\alpha}\textup{
\,and\,
}{\mathsf{F}}_{\rho}\colon{\mathbb{D}}_{e^{\rho}\cdot\alpha,{\mathrm{wt}_{0}}}\to{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\,.$

Indeed,

${\mathsf{F}}_{\rho}\left({\mathrm{ch}}_{0}(\gamma)-\int_{S}{\mathrm{ch}}(e^{\rho}\cdot\alpha)\gamma\right)={\mathrm{ch}}_{0}(e^{\rho}\gamma)-\int_{S}{\mathrm{ch}}(\alpha)e^{\rho}\gamma$

is zero in ${\mathbb{D}}_{\alpha}$.

####  1.2.3. Normalized classes in terms of descendent algebra

We can now finally explain how the classes $c_{k}(j)$ can be described in
terms of the descendent algebra. Let

$\delta=H/d\textup{ \;and\; }\rho=\left(\frac{3}{2}-\frac{\chi}{d}\right)H\,.$

Note that the class $\rho$ satisfies

$\int_{{\mathbb{P}}^{2}}{\mathrm{ch}}(\alpha)\cdot e^{\rho}=0\,.$

Consider the composition

${{\mathbb{D}}_{\alpha\cdot e^{\rho}}}$${{\mathbb{D}}_{\alpha\cdot
e^{\rho},{\mathrm{wt}_{0}}}}$${{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}}$${H^{*}(M_{d,\chi})\,.}$$\scriptstyle{\eta_{\delta}}$$\scriptstyle{{\mathsf{F}}_{\rho}}$$\scriptstyle{\xi}$

Then the normalized class $c_{k}(j)\in H^{*}(M_{d,\chi})$ can be obtained as
the image of ${\mathrm{ch}}_{k+j-1}(H^{j})\in{\mathbb{D}}_{\alpha\cdot
e^{\rho}}$ under the composition
$\xi\circ\mathsf{F}_{\rho}\circ\eta_{\delta}$.

To justify this, we check that the normalization (5) is satisfied. Indeed,
${\mathrm{ch}}_{0}(1)=0$ in ${\mathbb{D}}_{\alpha\cdot e^{\rho}}$ due to our
choice of $\rho$, and the composition annihilates ${\mathrm{ch}}_{1}(H)$ by
Proposition 1.12 (iv). We will also write $c_{k}(j)$ for the image of
${\mathrm{ch}}_{k+j-1}(H^{j})$ in ${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$,
and Proposition 1.13 identifies ${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$ with
a polynomial ring in infinitely many variables:

${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\cong{\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]\,.$

The variables are those in (4) with positive cohomological degrees, but
excluding $c_{1}(1)$. In particular, we can identify the ideal $I_{d,\chi}$
with an ideal in this polynomial ring.

When working with the moduli stacks ${\mathfrak{M}}_{d,\chi}$, we have a
canonical universal sheaf and there is no need to consider the
$\delta$-normalized sheaf. However, to get a description of tautological
classes and generators that is uniform under the isomorphism
${\mathfrak{M}}_{d,\chi}\cong{\mathfrak{M}}_{d,\chi+d}$, cf. Proposition 1.8
(ii), we shall still use the twist by ${\mathsf{F}}_{\rho}$ and denote by
$c_{k}(j)\in H^{*}({\mathfrak{M}}_{d,\chi})$ the image of
${\mathrm{ch}}_{k+j-1}(H^{j})$ after the composition

${{\mathbb{D}}_{\alpha\cdot
e^{\rho}}}$${{\mathbb{D}}_{\alpha}}$${H^{*}({\mathfrak{M}}_{d,\chi})\,.}$$\scriptstyle{{\mathsf{F}}_{\rho}}$$\scriptstyle{\xi}$

As in the case of moduli spaces, we use the isomorphism

${\mathbb{D}}_{\alpha}\cong{\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]$

to identify the ideal ${\mathfrak{I}}_{d,\chi}$ with an ideal of the above
polynomial ring in infinitely many variables. Note that we have an extra
variable $c_{1}(1)$ now and the map
$\eta\colon{\mathbb{D}}_{\alpha}\to{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$
becomes identified with the quotient map

(7)
${\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]\to{\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]$

that sets $c_{1}(1)$ to 0.

######  Remark 1.14.

In Section 6.1.2, we will use $\mathbb{D}_{\alpha,\mathrm{wt_{0}}}$ and the
identifications above to give a canonical (and equivalent) definition of the
Chern filtration on $H^{*}(M_{d,\chi})$. Previously, the Chern filtration is
only defined in [KPS23] using the normalized classes, which is very explicit
but conceptually less than ideal. We expect this new definition to shed more
light on the structure of the Chern filtration in various $P=C$ phenomena.

######  Remark 1.15.

By Proposition 1.8 (i), for $d\geq 3$ the second cohomology
$H^{2}(M_{d,\chi})$ has dimension 2 and basis $\\{c_{0}(2),c_{2}(0)\\}$, while
$H^{2}({\mathfrak{M}}_{d,\chi})$ has dimension 3 and basis
$\\{c_{0}(2),c_{1}(1),c_{2}(0)\\}$.

###  1.3. Descending tautological relations from the stacks.

When we write down and study tautological relations, it will be more natural
to do so for the moduli stacks where canonical universal sheaves exist. When
$\gcd(d,\chi)=1$, the good moduli map

$\pi\colon{\mathfrak{M}}_{d,\chi}\to M_{d,\chi}$

is a trivial ${\mathbb{G}}_{m}$-gerbe, so in particular we have an isomorphism

(8) $H^{*}({\mathfrak{M}}_{d,\chi})\cong H^{*}(M_{d,\chi}\times
B{\mathbb{G}}_{m})=H^{*}(M_{d,\chi})[u]\,$

where we write $u$ for the generator of $H^{2}(B{\mathbb{G}}_{m})$. However,
such an isomorphism is non-canonical as it depends on a choice of universal
sheaf ${\mathbb{F}}$ on $M_{d,\chi}\times\mathbb{P}^{2}$. Indeed, given
${\mathbb{F}}$, by the universal property of ${\mathfrak{M}}_{d,\chi}$ there
is a section $s_{\mathbb{F}}\colon M_{d,\chi}\to{\mathfrak{M}}_{d,\chi}$ such
that $s_{\mathbb{F}}^{\ast}{\mathcal{F}}={\mathbb{F}}$, and this section gives
a trivialization

$\mathrm{act}\circ(s_{\mathbb{F}}\times\textup{id}_{B{\mathbb{G}}_{m}})\colon
M_{d,\chi}\times B{\mathbb{G}}_{m}\to{\mathfrak{M}}_{d,\chi}\,$

where $\mathrm{act}$ is the $B{\mathbb{G}}_{m}$-action on
${\mathfrak{M}}_{d,\chi}$, cf. the proof of Proposition 1.5. In light of the
identifications explained above, it is natural to consider the isomorphism
associated to the $\delta$-normalized universal sheaf; even if
${\mathbb{F}}_{\delta}$ might not be represented by an actual universal sheaf,
it still induces an isomorphism (8) with rational coefficients.

######  Lemma 1.16.

Suppose that $\gcd(d,\chi)=1$. There is an isomorphism
$H^{*}({\mathfrak{M}}_{d,\chi})\cong H^{*}(M_{d,\chi})[u]$ that makes the
following diagram commute:

${{\mathbb{D}}_{\alpha}}$${{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}[u]}$${H^{*}({\mathfrak{M}}_{d,\chi})}$${H^{*}(M_{d,\chi})[u]}$$\scriptstyle{\varphi}$

In particular, we have
$I_{d,\chi}=\eta\big{(}{\mathfrak{I}}_{d,\chi}\big{)}\,$, or more explicitly

${\mathfrak{I}}_{d,\chi}|_{c_{1}(1)=0}=I_{d,\chi}$

via the identification (7).

###### Proof.

The idea of the proof is essentially the same as in [Joy18, Proposition 3.24],
so we will just sketch it. Let ${\mathbb{F}}$ be any universal sheaf; there is
an $m\in{\mathbb{Z}}_{+}$ and a $B{\mathbb{Z}}_{m}$-fibration $f\colon T\to
M_{d,\chi}$ such that $T$ has a line bundle $L$ with
$c_{1}(L)=-f^{\ast}\xi_{{\mathbb{F}}}\big{(}{\mathrm{ch}}_{1}(\delta)\big{)}.$
Then $f^{\ast}{\mathbb{F}}\otimes L$ is a sheaf on $T\times\mathbb{P}^{2}$
with the property that

${\mathrm{ch}}(f^{\ast}{\mathbb{F}}\otimes
L)=f^{\ast}{\mathrm{ch}}({\mathbb{F}}_{\delta})\,.$

By the universal property, we get a map $s\colon T\to{\mathfrak{M}}_{d,\chi}$.
A $B{\mathbb{Z}}_{m}$-fibration induces an isomorphism $f^{\ast}$ on rational
cohomology. Thus, we obtain the required isomorphism by composing

$H^{*}({\mathfrak{M}}_{d,\chi})\xlongrightarrow{\mathrm{act}^{\ast}}H^{*}({\mathfrak{M}}_{d,\chi}\times
B{\mathbb{G}}_{m})\xlongrightarrow{s^{\ast}\otimes\textup{id}}H^{*}(T\times
B{\mathbb{G}}_{m})\xlongleftarrow{f^{\ast}\otimes\textup{id}}H^{*}(M_{d,\chi}\times
B{\mathbb{G}}_{m})\,.$

The diagram is easily shown to commute using Proposition 1.12. The last
statement follows from the fact that $\eta$ is obtained by applying $\varphi$
and then setting $u\mapsto 0$. ∎

##  2\. Geometric relations

We will now focus on proving three families of relations — we call these
families Mumford relations (${\mathsf{MR}}$), generalized Mumford relations
(${\mathsf{GMR}}$) and base relations (${\mathsf{BR}}$) — among the generators
of the descendent algebra. In other words, we will construct explicit ideals
${\mathfrak{I}}^{{\mathsf{MR}}}_{d,\chi}$,
${\mathfrak{I}}^{{\mathsf{GMR}}}_{d,\chi}$ and
${\mathfrak{I}}^{{\mathsf{BR}}}_{d,\chi}$, and show that

${\mathfrak{I}}^{{\mathsf{MR}}}_{d,\chi},{\mathfrak{I}}^{{\mathsf{GMR}}}_{d,\chi},{\mathfrak{I}}^{{\mathsf{BR}}}_{d,\chi}\subset{\mathfrak{I}}_{d,\chi}\,.$

We will denote by ${\mathfrak{I}}^{\mathsf{Geom}}_{d,\chi}$ the ideal
generated by
${\mathfrak{I}}^{{\mathsf{MR}}}_{d,\chi},{\mathfrak{I}}^{{\mathsf{GMR}}}_{d,\chi},{\mathfrak{I}}^{{\mathsf{BR}}}_{d,\chi}$.
As we will explain, the determination of
${\mathfrak{I}}^{{\mathsf{GMR}}}_{d,\chi}$ requires knowledge of the
cohomology rings of ${\mathfrak{M}}_{d^{\prime},\chi^{\prime}}$ for ‘smaller’
values $(d^{\prime},\chi^{\prime})$, leading to an inductive approach to
understanding this ideal and more generally the cohomology ring structure.

Having defined these ideals, the natural question is the following (cf.
Question 0.5):

######  Question 2.1.

Do we have ${\mathfrak{I}}^{\mathsf{Geom}}_{d,\chi}={\mathfrak{I}}_{d,\chi}$?

If the answer to this question were positive, we would have a recursive
algorithm to compute the cohomology rings of ${\mathfrak{M}}_{d,\chi}$. We
will look into Question 2.1 in Section 5.2 for small $d$; it turns out that
the answer is positive for $d\leq 4$ and $(d,\chi)=(5,2)$, but negative for
$(d,\chi)=(5,1)$.

We work at the level of stacks where universal sheaves exist, but this also
gives relations for the good moduli spaces $M_{d,\chi}$ when $\gcd(d,\chi)=1$
by Lemma 1.16. More precisely, if we define

$I^{\bullet}_{d,\chi}=\eta({\mathfrak{I}}^{\bullet}_{d,\chi})\quad\textup{for
\,}\bullet={\mathsf{MR}},{\mathsf{GMR}},{\mathsf{BR}},{\mathsf{Geom}}$

then we have $I^{\mathsf{Geom}}_{d,\chi}\subset I_{d,\chi}$.

###  2.1. Mumford relations

As explained in Section 0.2, one general approach to proving geometric
relations on the cohomology of moduli spaces is to construct some vector
bundle ${\mathcal{V}}$ out of the moduli data (in our case, the universal
sheaf ${\mathbb{F}}$) with a rank $r$ that we can compute by
Hirzebruch–Riemann–Roch; then the vanishing of Chern classes

$c_{j}({\mathcal{V}})=0\quad\textup{for }j>r\,$

may give non-trivial relations. Usually ${\mathcal{V}}$ is first constructed
as a complex, and then shown to be a vector bundle by some cohomological
vanishing. Mumford relations follow this strategy applied to the following
vanishing result:

######  Proposition 2.2.

Let $F$ be a semistable sheaf of type $(d,\chi)$. Assume

$\chi\geq g_{d}\coloneqq\frac{(d-1)(d-2)}{2},$

then $H^{1}(F)=H^{2}(F)=0$.

###### Proof.

This is proven in [DM11, Proposition 2.1.3], but we provide a proof for
completeness. The vanishing $H^{2}(F)=0$ is obvious since $F$ is one-
dimensional. Let $C^{\prime}$ be the scheme-theoretic support of $F$, thus
$0<d^{\prime}:=\deg(C^{\prime})\leq d$. Suppose that $H^{1}(F)\neq 0$, we can
pick an element $s\neq 0$ in the dual

$H^{1}(C^{\prime},F)^{\vee}=\mathrm{Hom}(F,\omega_{C^{\prime}})=\mathrm{Hom}(F,\mathcal{O}_{C^{\prime}}(-3+d^{\prime})).$

This gives rise to a non-zero morphism

$s:F\to\mathcal{O}_{C^{\prime}}(-3+d^{\prime}),$

whose image we denote by $G(-3+d^{\prime})$ with
$G\subset\mathcal{O}_{C^{\prime}}$. By the semistability of $F$, we have

(9) $\frac{\chi}{d}\leq\frac{\chi(G(-3+d^{\prime}))}{\deg
G}=\frac{\chi(G)}{\deg G}-3+d^{\prime}.$

To estimate the RHS, we consider the diagram

${T}$${0}$${G}$${{\mathcal{O}_{C^{\prime}}}}$${{\mathcal{O}_{C^{\prime\prime}}}}$${0}$${0}$${{\widetilde{G}}}$${{\mathcal{O}_{C^{\prime}}}}$${{\mathcal{O}_{C_{\mathrm{pure}}^{\prime\prime}}}}$${{0.}}$

Here $C^{\prime\prime}$ is a closed subvariety of $C^{\prime}$, and
$\mathcal{O}_{C^{\prime\prime}_{\mathrm{pure}}}$ is the quotient of
$\mathcal{O}_{C^{\prime\prime}}$ by its torsion part $T$. Thus it is the
structure sheaf of a planar curve of degree $0\leq
d^{\prime\prime}<d^{\prime}$. We have

$\frac{\chi(G)}{\deg G}\leq\frac{\chi(\widetilde{G})}{\deg
G}=\frac{\chi(\mathcal{O}_{C^{\prime}})-\chi(\mathcal{O}_{C^{\prime\prime}_{\mathrm{pure}}})}{d^{\prime}-d^{\prime\prime}}.$

Using the formula $\chi(\mathcal{O}_{C_{d}})=1-g_{d}=1-\frac{(d-1)(d-2)}{2}$
for any degree $d$ planar curve, an elementary computation shows that

$\max_{0\leq d^{\prime\prime}<d^{\prime}\leq
d}\left(\frac{\chi(\mathcal{O}_{C^{\prime}})-\chi(\mathcal{O}_{C^{\prime\prime}_{\mathrm{pure}}})}{d^{\prime}-d^{\prime\prime}}-3+d^{\prime}\right)=\frac{d-3}{2}.$

It follows from (9) that $\chi\leq\frac{d(d-3)}{2}$, but this contradicts our
assumption $\chi\geq\frac{(d-1)(d-2)}{2}$. ∎

By the cohomology and base change theorem, it follows that for $(d,\chi)$ with
$\chi\geq g$ the complex $Rp_{\ast}{\mathcal{F}}\in
D^{b}({\mathfrak{M}}_{d,\chi})$ is actually a vector bundle with fibers
$H^{0}(F)$ over $[F]\in{\mathfrak{M}}_{d,\chi}$. Moreover, the rank of this
vector bundle is $\chi$. It follows that $c_{j}(Rp_{\ast}{\mathcal{F}})=0$ for
every $j>\chi\geq g$. To write down this formula in terms of descendents,
Grothendieck–Riemann–Roch yields

${\mathrm{ch}}(Rp_{\ast}{\mathcal{F}})=p_{\ast}\big{(}{\mathrm{ch}}({\mathcal{F}}){\mathrm{td}}(T_{p})\big{)}=p_{\ast}\big{(}{\mathrm{ch}}({\mathcal{F}})q^{\ast}{\mathrm{td}}({\mathbb{P}}^{2})\big{)}\,,$

so ${\mathrm{ch}}_{j}(Rp_{\ast}{\mathcal{F}})$ is given by the geometrical
realization of ${\mathrm{ch}}_{j}({\mathrm{td}}({\mathbb{P}}^{2}))$. Thus by
Newton’s identity, the total Chern class $c(Rp_{\ast}{\mathcal{F}})$ is the
geometric realization of

$\exp\left(\sum_{j\geq
1}(-1)^{j-1}(j-1)!{\mathrm{ch}}_{j}({\mathrm{td}}({\mathbb{P}}^{2}))\right).$

The upshot of the above discussion is the following:

######  Proposition 2.3.

Suppose that $j>\chi\geq g$. Then

$\left[\exp\left(\sum_{j\geq
1}(-1)^{j-1}(j-1)!{\mathrm{ch}}_{j}({\mathrm{td}}(\mathbb{P}^{2}))\right)\right]_{2j}\in{\mathfrak{I}}_{d,\chi}\,.$

For a fixed $(d,\chi)$, we can use the isomorphisms
${\mathfrak{M}}_{d,\chi}\cong{\mathfrak{M}}_{d,\pm\chi+kd}$ to obtain
relations by Proposition 2.3 if we take sufficiently large $k$. Given an
arbitrary $(d,\chi)$, a choice of signs $\pm$ and $j,k\in\mathbb{Z}$ such that
$j>\pm\chi+kd\geq g$, we get a relation

${\mathsf{MR}}_{d,\chi}^{\pm,k,j}\in{\mathfrak{I}}_{d,\chi}$

which corresponds to

$\left[\exp\left(\sum_{k\geq
1}(-1)^{k-1}(k-1)!{\mathrm{ch}}_{k}({\mathrm{td}}({\mathbb{P}}^{2}))\right)\right]_{2j}\in{\mathfrak{I}}_{d,\pm\chi+kd}\,.$

under the ismomorphism
${\mathfrak{I}}_{d,\chi}\cong{\mathfrak{I}}_{d,\pm\chi+kd}$.

We define the ideal

${\mathfrak{I}}^{\mathsf{MR}}_{d,\chi}:=\big{\langle}{\mathsf{MR}}_{d,\chi}^{\pm,k,j}\big{|}\,j>\pm\chi+kd\geq
g\big{\rangle}\subset{\mathfrak{I}}_{d,\chi}\,.$

######  Remark 2.4.

The relations of the form ${\mathsf{MR}}_{d,\chi}^{-,k,j}$ can also be
understood as coming from the dual version of Proposition 2.2, namely

$H^{0}(F)=H^{2}(F)=0\quad\textup{if }\chi\leq-g.$

######  Example 2.5.

Recall that ${\mathfrak{M}}_{1,0}$ is a ${\mathbb{G}}_{m}$-gerbe over
$M_{1,0}\simeq\check{{\mathbb{P}}}^{2}$ where $\check{{\mathbb{P}}}^{2}$ is
the dual projective plane. Denote the dual hyperplane section by $\check{H}$.
Then we have

(10) $H^{*}({\mathfrak{M}}_{1,0},{\mathbb{Q}})\simeq H^{*}(M_{1,0})\otimes
H^{*}(B{\mathbb{G}}_{m})={\mathbb{Q}}[{\check{H}}]/{\check{H}}^{3}\otimes{\mathbb{Q}}[u].$

We work this out using Mumford relations. Write

$\gamma_{k}=\mathrm{td}({\mathbb{P}}^{2})\cdot
e^{kH}\;\;\textrm{and}\;\;{\mathcal{V}}_{k}=Rp_{*}\left({\mathcal{F}}\otimes
q^{*}{\mathcal{O}}_{{\mathbb{P}}^{2}}(k)\right),\;\;\;\textrm{for
}k\in{\mathbb{Z}}.$

By the above discussions, we have

$\displaystyle c_{j}(-{\mathcal{V}}_{-1})=0\quad\textrm{for }j>1,\quad
c_{j}({\mathcal{V}}_{0})=0\quad\textrm{for }j>0,$ $\displaystyle
c_{j}({\mathcal{V}}_{1})=0\quad\textrm{for }j>1,\quad
c_{j}({\mathcal{V}}_{2})=0\quad\textrm{for }j>2.$

Set further $\alpha={\mathrm{ch}}_{1}(\gamma_{1})$ and
$\beta={\mathrm{ch}}_{1}(\gamma_{-1})$. The second vanishing above implies
$\mathrm{ch}_{k}(\gamma_{0})=0$, and the first and third imply

(11)
${\mathrm{ch}}_{k}(\gamma_{1})=\frac{1}{k!}\alpha^{k},\quad{\mathrm{ch}}_{k}(\gamma_{-1})=\frac{(-1)^{k+1}}{k!}\beta^{k}\quad\textrm{for
}k\geq 1.$

Since $\\{\gamma_{-1},\gamma_{0},\gamma_{1}\\}$ form a basis of
$H^{*}({\mathbb{P}}^{2})$, it follows from Theorem 1.3 that $\alpha,\beta$
generate $H^{*}({\mathfrak{M}}_{1,0})$. Finally, the last vanishing gives

(12)
$c_{3}({\mathcal{V}}_{2})=\frac{1}{6}{\mathrm{ch}}_{1}(\gamma_{2})^{3}-{\mathrm{ch}}_{1}(\gamma_{2}){\mathrm{ch}}_{2}(\gamma_{2})+2{\mathrm{ch}}_{3}(\gamma_{2})=0.$

Using $\gamma_{2}=3\gamma_{1}+\gamma_{-1}-3\gamma_{0}$ and (11), the relation
(12) simplifies into $(\alpha+\beta)^{3}=0$. Thus

$H^{*}({\mathfrak{M}}_{1,0},{\mathbb{Q}})={\mathbb{Q}}[\alpha,\beta]/(\alpha+\beta)^{3},$

matching the presentation (10). Note that

$\alpha+\beta={\mathrm{ch}}_{1}(\gamma_{1}+\gamma_{-1})={\mathrm{ch}}_{1}(\gamma_{1}+\gamma_{-1}-2\gamma_{0})={\mathrm{ch}}_{1}(H^{2})=c_{0}(2)$

is indeed the (pull-back of) hyperplane class $\check{H}$, cf. Proposition 1.8
(i).

###  2.2. Generalized Mumford relations

Following the same strategy but using cohomological vanishing provided by
stability, we can construct a generalized version of Mumford relations.

######  Proposition 2.6.

Let $F,F^{\prime}$ be semistable sheaves on $\mathbb{P}^{2}$ of topological
types $(d,\chi)$ and $(d^{\prime},\chi^{\prime})$, respectively. Suppose that

(13)
$\frac{\chi^{\prime}}{d^{\prime}}<\frac{\chi}{d}<\frac{\chi^{\prime}}{d^{\prime}}+3\,.$

Then we have

$\operatorname{Hom}(F,F^{\prime})=\operatorname{Ext}^{2}(F,F^{\prime})=0\,.$

###### Proof.

The vanishing $\mathrm{Hom}(F,F^{\prime})=0$ follows from the first inequality
in (13) and the semistability of $F$ and $F^{\prime}$. To see that
$\mathrm{Ext}^{2}(F,F^{\prime})=0$, we use Serre duality

$\mathrm{Ext}^{2}(F,F^{\prime})\simeq\mathrm{Hom}(F^{\prime},F\otimes\mathcal{O}_{\mathbb{P}^{2}}(-3))^{\vee},$

which vanishes due to the second inequality in (13). ∎

Thanks to Proposition 2.6, whenever $(d,\chi)$ and
$(d^{\prime},\chi^{\prime})$ satisfy the inequality (13), we can construct a
vector bundle on the product
${\mathfrak{M}}_{d,\chi}\times{\mathfrak{M}}_{d^{\prime},\chi^{\prime}}$ of
the moduli stacks . Indeed, let ${\mathcal{F}}$ and ${\mathcal{F}}^{\prime}$
be the universal sheaves on ${\mathfrak{M}}_{d,\chi}\times{\mathbb{P}}^{2}$
and ${\mathfrak{M}}_{d^{\prime},\chi^{\prime}}\times{\mathbb{P}}^{2}$,
respectively. Then the complex

${\mathcal{V}}:=R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}},{\mathcal{F}}^{\prime})[1]$

is a vector bundle, where
$p\colon{\mathfrak{M}}_{d,\chi}\times{\mathfrak{M}}_{d^{\prime},\chi^{\prime}}\times{\mathbb{P}}^{2}\to{\mathfrak{M}}_{d,\chi}\times{\mathfrak{M}}_{d^{\prime},\chi^{\prime}}$
and we omit the pull-backs of ${\mathcal{F}},{\mathcal{F}}^{\prime}$ to the
triple product. This vector bundle has fiber
$\operatorname{Ext}^{1}(F,F^{\prime})$ over a point
$([F],[F^{\prime}])\in{\mathfrak{M}}_{d,\chi}\times{\mathfrak{M}}_{d^{\prime},\chi^{\prime}}$,
and its rank is equal to

$\dim\mathrm{Ext}^{1}(F,F^{\prime})=-\chi(F,F^{\prime})=d\cdot d^{\prime}\,.$

Thus we obtain a relation

$c_{j}(-R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}},{\mathcal{F}}^{\prime}))=0\quad\textup{for
}j>dd^{\prime}\,$

which holds in
$H^{*}({\mathfrak{M}}_{d,\chi}\times{\mathfrak{M}}_{d^{\prime},\chi^{\prime}})$.
To extract from this a relation on ${\mathfrak{M}}_{d,\chi}$, we take a
homology class $A\in H_{*}({\mathfrak{M}}_{d^{\prime},\chi^{\prime}})$ and
integrate along it, i.e. we apply the slant product map

$\int_{A}\colon
H^{*}({\mathfrak{M}}_{d,\chi}\times{\mathfrak{M}}_{d^{\prime},\chi^{\prime}})\to
H^{*}({\mathfrak{M}}_{d,\chi})\,.$

To describe explicitly the relations in terms of descendents, we use
Grothendieck–Riemann–Roch again to find a representative of
$-R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}},{\mathcal{F}}^{\prime})$
in the descendent algebra. For this, we consider the realization map

${\mathbb{D}}\otimes{\mathbb{D}}\to H^{*}({\mathfrak{M}}_{d,\chi})\otimes
H^{*}({\mathfrak{M}}_{d^{\prime},\chi^{\prime}})\,.$

Define $C$ by

$C=\exp\left(\sum_{\begin{subarray}{c}a,b\geq 0\\\
(a,b)\neq(0,0)\end{subarray}}\sum_{i}(-1)^{b+d_{i}^{L}}(a+b-1)!{\mathrm{ch}}_{a}(\gamma_{i}^{L})\otimes{\mathrm{ch}}_{b}(\gamma_{i}^{R})\right)\in{\mathbb{D}}\otimes{\mathbb{D}}$

where

(14)
$\sum_{i}\gamma_{i}^{L}\otimes\gamma_{i}^{R}=\Delta_{\ast}{\mathrm{td}}({\mathbb{P}}^{2})$

is the Künneth decomposition of the push-forward of
${\mathrm{td}}({\mathbb{P}}^{2})$ along the diagonal and $\gamma_{i}^{L}\in
H^{2d_{i}^{L}}({\mathbb{P}}^{2})$. More concretely, we take the decomposition

$\Delta_{\ast}{\mathrm{td}}({\mathbb{P}}^{2})=H^{2}\otimes 1+H\otimes
H+1\otimes H^{2}+\frac{3}{2}H\otimes H^{2}+\frac{3}{2}H^{2}\otimes
H+H^{2}\otimes H^{2}\,.$

We let $C_{j}$ be the part of $C$ with cohomological degree $2j$ (recall that
by definition ${\mathrm{ch}}_{a}(\gamma)$ has cohomological degree $2a$).

######  Lemma 2.7.

The geometric realization of $C_{j}$ equals
$c_{j}(-R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}},{\mathcal{F}}^{\prime}))$.

###### Proof.

This is a standard application of Grothendieck–Riemann–Roch, a diagonal trick
and Newton’s identity. We include a sketch of the proof for completeness, see
[She16, Section 3.1] for details. By Grothendieck–Riemann–Roch, we have

$\mathrm{ch}(R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}},{\mathcal{F}}^{\prime}))=p_{*}(\mathrm{ch}^{\vee}(\mathcal{F}^{\prime})\cdot\mathrm{ch}(\mathcal{F})\cdot
q^{*}{\mathrm{td}}({\mathbb{P}}^{2})).$

Consider the following commutative diagram, where
$\delta=\textup{id}\times\Delta$ and $p=\pi_{12}\circ\delta$:

${{\mathfrak{M}_{d,\chi}\times\mathfrak{M}_{d^{\prime},\chi^{\prime}}\times\mathbb{P}^{2}}}$${{\mathfrak{M}_{d,\chi}\times\mathfrak{M}_{d^{\prime},\chi^{\prime}}\times\mathbb{P}^{2}\times\mathbb{P}^{2}}}$${{\mathfrak{M}_{d,\chi}\times\mathfrak{M}_{d^{\prime},\chi^{\prime}}}}$${{\mathbb{P}^{2}}}$${{\mathbb{P}^{2}\times\mathbb{P}^{2}.}}$$\scriptstyle{\textup{id}\times\Delta}$$\scriptstyle{\pi_{12}}$$\scriptstyle{q}$$\scriptstyle{\pi_{34}}$$\scriptstyle{\Delta}$

By projection formula and the Künneth decomposition (14), we have

$\displaystyle
p_{*}(\mathrm{ch}^{\vee}(\mathcal{F}^{\prime})\cdot\mathrm{ch}(\mathcal{F})\cdot
q^{*}{\mathrm{td}}({\mathbb{P}}^{2}))$
$\displaystyle=\sum_{i}\pi_{12,*}\left(\mathrm{ch}^{\vee}(\mathcal{F}^{\prime})\cdot\mathrm{ch}(\mathcal{F})\cdot\pi_{34}^{*}(\gamma_{i}^{L}\otimes\gamma_{i}^{R})\right)$
$\displaystyle=\sum_{i}\sum_{a,b\geq
0}(-1)^{a+2-d_{i}^{L}}{\mathrm{ch}}_{a}(\gamma_{i}^{L})\otimes{\mathrm{ch}}_{b}(\gamma_{i}^{R}).$

The sign in the last equality is obtained by tracing back the degree of
${\mathrm{ch}}^{\vee}({\mathcal{F}}^{\prime})$ used for
${\mathrm{ch}}_{a}(\gamma_{i}^{L})$. The lemma follows by applying Newton’s
identity as in Section 2.1. ∎

Given $\alpha=(d,\chi),\,\alpha^{\prime}=(d^{\prime},\chi^{\prime})$
satisfying (13), $j>dd^{\prime}$ and $A\in
H_{*}({\mathfrak{M}}_{d^{\prime},\chi^{\prime}})$, we define the relation
${\mathsf{GMR}}_{\alpha}^{\alpha^{\prime},j,A}\in{\mathbb{D}}_{\alpha}$ to be
the image of $C_{j}$ under

${\mathbb{D}}\otimes{\mathbb{D}}\to{\mathbb{D}}_{\alpha}\otimes{\mathbb{D}}_{\alpha^{\prime}}\xlongrightarrow{\textup{id}\otimes\xi}{\mathbb{D}}_{\alpha}\otimes
H^{*}({\mathfrak{M}}_{d^{\prime},\chi^{\prime}})\xlongrightarrow{\int_{A}}{\mathbb{D}}_{\alpha}\,.$

The upshot of the above discussion is the following:

######  Proposition 2.8.

For every $\alpha=(d,\chi),\,\alpha^{\prime}=(d^{\prime},\chi^{\prime})$
satisfying (13), $j>dd^{\prime}$ and $A\in
H_{*}({\mathfrak{M}}_{d^{\prime},\chi^{\prime}})$, we obtain a tautological
relation
${\mathsf{GMR}}_{\alpha}^{\alpha^{\prime},j,A}\in{\mathfrak{I}}_{d,\chi}$.

Thus, we define the ideals

${\mathfrak{I}}^{{\mathsf{GMR}},\alpha^{\prime}}_{\alpha}=\Big{\langle}{\mathsf{GMR}}_{\alpha}^{\alpha^{\prime},j,A}\big{|}\,j>dd^{\prime},\,A\in
H_{*}({\mathfrak{M}}_{d^{\prime},\chi^{\prime}})\Big{\rangle}\subset{\mathfrak{I}}_{d,\chi}\,.$

######  Remark 2.9.

There are several seemingly different ways to produce generalized Mumford
relations. First, one can easily check that
${\mathfrak{I}}^{{\mathsf{GMR}},\alpha^{\prime}}_{\alpha}={\mathfrak{I}}^{{\mathsf{GMR}},d^{\prime},\chi^{\prime}+kd^{\prime}}_{d,\chi+kd},$
so the isomorphisms ${\mathfrak{M}}_{d,\chi}\cong{\mathfrak{M}}_{d,\chi+kd}$
do not give new relations. One can also use the isomorphism
${\mathfrak{M}}_{d,\chi}\cong{\mathfrak{M}}_{d,-\chi}$ and look at
$(d^{\prime},\chi^{\prime})$ satisfying the dual version of the inequality
(13). A third way is to use the vector bundle
$R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}}^{\prime},{\mathcal{F}})$
instead of
$R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}},{\mathcal{F}}^{\prime})$.
The latter two approaches are actually equivalent by an application of
Grothendieck–Verdier duality [Huy06, Section 3.4]. Nevertheless, one can show
that they also do not produce new relations.

######  Remark 2.10.

In the case $\gcd(d^{\prime},\chi^{\prime})=1$, it is often more efficient to
use the good moduli space $M_{d^{\prime},\chi^{\prime}}$ to produce relations.
We use $(-)^{\dagger}$ to denote the dual of an operator. Given
$\overline{A}\in H^{*}(M_{d^{\prime},\chi^{\prime}})$, we can consider the
relation
${\mathsf{GMR}}_{\alpha}^{\alpha^{\prime},j,\overline{A}}\coloneqq{\mathsf{GMR}}_{\alpha}^{\alpha^{\prime},j,A}$
where $A=\eta^{\dagger}([M_{d^{\prime},\chi^{\prime}}]\cap\overline{A})$. By
Lemma A.2, we have

${\mathsf{GMR}}^{\alpha^{\prime},j,\mathsf{R}_{-1}^{\dagger}(A)}_{\alpha}=(j-1-dd^{\prime}){\mathsf{GMR}}^{\alpha^{\prime},j-1,A}_{\alpha}.$

From this identity and Lemma 1.16, we conclude that the relations
${\mathsf{GMR}}_{\alpha}^{\alpha^{\prime},j,a}$ generate
${\mathfrak{I}}_{\alpha}^{{\mathsf{GMR}},\alpha^{\prime}}$. In practice, we
will indeed use the good moduli space $M_{d^{\prime},\chi^{\prime}}$ when
possible, for example in the computations of Section 5.2.

Because the construction of the ideal
${\mathfrak{I}}^{{\mathsf{GMR}},\alpha^{\prime}}_{\alpha}$ requires
integration against homology classes of
${\mathfrak{M}}_{d^{\prime},\chi^{\prime}}$, to explicitly compute this ideal
we need to understand $H^{*}({\mathfrak{M}}_{d^{\prime},\chi^{\prime}})$
first. Thus it makes sense to determine the cohomology ring structure
inductively for all $(d,\chi)$, by using the geometric relations with
$d^{\prime}<d$. It turns out that, when working with moduli stacks where
$\mathrm{gcd}(d,\chi)>1$, it is also useful to take $d^{\prime}=d$ and
$\chi^{\prime}$ with a smaller $\gcd$.888For example, the computation of
$H^{*}({\mathfrak{M}}_{3,0})$ in Section 5.2.2 requires $\mathsf{GMR}$ from
$M_{3,1}$ and $M_{3,2}$. This motivates

######  Definition 2.11.

We say that $(d^{\prime},n^{\prime})\prec(d,n)$ if either $d^{\prime}<d$, or
$d^{\prime}=d$ and $\gcd(d^{\prime},\chi^{\prime})<\gcd(d,\chi)$. We define
the ideal of generalized Mumford relations to be

${\mathfrak{I}}^{\mathsf{GMR}}_{d,\chi}:=\sum_{\begin{subarray}{c}(d^{\prime},\chi^{\prime})\prec(d,\chi)\\\
\textrm{satisfying }\eqref{eq:
inequalitygmr}\end{subarray}}{\mathfrak{I}}^{{\mathsf{GMR}},\alpha^{\prime}}_{\alpha}\subset{\mathfrak{I}}_{d,\chi}.$

######  Remark 2.12.

As pointed out in Section 5.2.5, the ideal
${\mathfrak{I}}^{\mathsf{Geom}}_{5,1}$ turns out to be strictly smaller than
${\mathfrak{I}}_{5,1}$. On the other hand, we have
${\mathfrak{I}}^{\mathsf{Geom}}_{5,2}={\mathfrak{I}}_{5,2}$. Thus from an
algorithmic point of view, it could make sense to extend $\prec$ so that
$(5,2)\prec(5,1)$ and include ${\mathfrak{I}}^{{\mathsf{GMR}},5,2}_{5,1}$ in
the geometric relations. We have not been able to check if this produces the
missing relations in ${\mathfrak{I}}^{\mathsf{Geom}}_{5,1}$ due to the
computational complexity involved.

######  Remark 2.13.

On the moduli space level, the generalized Mumford relations
$I_{d,\chi}^{{\mathsf{GMR}},1,\chi^{\prime}}$ already played a fundamental
role in some recent developments understanding the cohomology rings. For
example, one key step in Theorem 1.9 (i) consists of showing that these
relations express all (normalized) tautological classes in terms of the $3d-7$
generators, and Theorem 1.7 follows from a careful study of the three
relations in Theorem 1.9 (ii), which can be produced by
$I_{d,\chi}^{{\mathsf{GMR}},1,\chi^{\prime}}$. The current paper builds on
these progresses and explore the full power of generalized Mumford relations;
indeed, they make up the majority of all geometric relations we formulate.

###  2.3. Base relations

The set of relations that we discuss now have a different nature; they use the
Hilbert–Chow morphism $h\colon{\mathfrak{M}}_{d,\chi}\to|d\cdot H|$ in a
crucial way. Recall that $|d\cdot H|={\mathbb{P}}^{b}$ where $b=d(d+3)/2$ and
that ${\mathrm{ch}}_{1}(H^{2})=c_{0}(2)$ is the pull-back of the hyperplane
class via $h$. It follows that $\mathrm{ch}_{1}(H^{2})^{b+1}=0$. More
generally, the following is true:

######  Proposition 2.14.

For any $i_{1},\ldots,i_{b+1}\geq 1$,

${\mathrm{ch}}_{i_{1}}(H^{2})\cdots{\mathrm{ch}}_{i_{b+1}}(H^{2})\in{\mathfrak{I}}_{d,\chi}$

is a tautological relation.

###### Proof.

By the Grothendieck–Riemann–Roch theorem, the Chern character
$\mathrm{ch}({\mathcal{F}})$ is a cycle pushed forward from
$\mathrm{supp}({\mathcal{F}})\subset{\mathfrak{M}}_{d,\chi}\times\mathbb{P}^{2}$.
Recall the geometric realization

${\mathrm{ch}}_{i}(H^{2})=\Big{[}p_{\ast}\big{(}{\mathrm{ch}}({\mathcal{F}})\cdot
q^{\ast}H^{2})\Big{]}_{2i}\in H^{2i}({\mathfrak{M}}_{d,\chi}).$

If we pick a point ${\mathsf{p}}\in\mathbb{P}^{2}$ to represent the class
$H^{2}$, then $q^{*}H^{2}$ can be identified with
${\mathfrak{M}}_{d,\chi}\times\\{{\mathsf{p}}\\}$, and the class
$\mathrm{ch}_{i}(H^{2})$ is a push-forward from the intersection
$\mathrm{supp}(\mathcal{F})\cap({\mathfrak{M}}_{d,\chi}\times\\{{\mathsf{p}}\\})$.
In particular, a sheaf
$[F]\in[\mathrm{ch}_{i}(H^{2})]\subset{\mathfrak{M}}_{d,\chi}$ must be
supported at the point ${\mathsf{p}}$. Now we pick $b+1$ points
${\mathsf{p}}_{1},{\mathsf{p}}_{2},\ldots,{\mathsf{p}}_{b+1}\in\mathbb{P}^{2}$
in general position, then there are no degree $d$ curves passing through all
of them. It follows that

${\mathrm{ch}}_{i_{1}}(H^{2})\cdots{\mathrm{ch}}_{i_{b+1}}(H^{2})=0.\qed$

We define the ideal of base relations

${\mathfrak{I}}^{\mathsf{BR}}_{d,\chi}:=\left\langle{\mathrm{ch}}_{i_{1}}(H^{2})\cdots{\mathrm{ch}}_{i_{b+1}}(H^{2})\mid
i_{1},\ldots,i_{b+1}\geq 1\right\rangle\subset{\mathfrak{I}}_{d,\chi}.$

Note that the base relations all have cohomological degrees $\geq 2(b+1)$.

##  3\. Virasoro representations

In this section, we introduce the Virasoro operators and explain how they can
be used to study the cohomology ring of the moduli stacks and spaces. We
explain the Virasoro constraints for moduli spaces $M_{d,\chi}$ with
$\gcd(d,\chi)=1$ proven in [LM24] and an important consequence: the ideal of
tautological relations is preserved by the Virasoro operators, hence inducing
a representation of $\mathsf{Vir}_{\geq-1}$ on
$H^{*}({\mathfrak{M}}_{d,\chi})$. In fact, such a property is proven in loc.
cit. for any ${\mathfrak{M}}_{d,\chi}$ with $(d,\chi)$ not necessarily
coprime. Second, we refine the preservation of the ideal under the Virasoro
operators to the ideal of each type
${\mathsf{MR}}$/${\mathsf{GMR}}$/${\mathsf{BR}}$/${\mathsf{Geom}}$.
Furthermore, we prove that each ideal is generated by suitable primitive
elements under the Virasoro operators. Working with the Virasoro operators and
primitive relations is computationally much more effective than working
directly with the entire ideals.

###  3.1. Virasoro operators and Virasoro constraints

Recall the definition of the descendent algebra ${\mathbb{D}}$ from Section
1.1. The Virasoro operators $\\{{\mathsf{L}}_{n}\\}_{n\geq-1}$ acting on
${\mathbb{D}}$ is defined as a sum
${\mathsf{L}}_{n}={\mathsf{R}}_{n}+{\mathsf{T}}_{n}$ of the two operators:
${\mathsf{R}}_{n}$ is a derivation operator such that

${\mathsf{R}}_{n}({\mathrm{ch}}_{i}(\gamma)):=i(i+1)\cdots(i+n){\mathrm{ch}}_{i+n}(\gamma)$

and ${\mathsf{T}}_{n}$ is a multiplication operator by the element

${\mathsf{T}}_{n}:=\sum_{a+b=n,\,a,b\geq
0}\sum_{i}a!b!\,(-1)^{2-d_{i}^{L}}{\mathrm{ch}}_{a}(\gamma_{i}^{L})\cdot{\mathrm{ch}}_{b}(\gamma_{i}^{R}),$

where $\gamma_{i}^{L}$ and $\gamma_{i}^{R}$ are the same as in (14). Note that
${\mathsf{L}}_{-1}={\mathsf{R}}_{-1}$ agrees with Definition 1.2. One can
check that the operators satisfy the Virasoro bracket relations

$[{\mathsf{R}}_{n},{\mathsf{R}}_{m}]=(m-n){\mathsf{R}}_{n+m},\quad[{\mathsf{L}}_{n},{\mathsf{L}}_{m}]=(m-n){\mathsf{L}}_{n+m}.$

The weight zero Virasoro operator is defined as a combination of the Virasoro
operators

${\mathsf{L}}_{{\mathrm{wt}_{0}}}:=\sum_{n\geq-1}\frac{(-1)^{n}}{(n+1)!}{\mathsf{L}}_{n}\circ{\mathsf{L}}_{-1}^{n+1}:{\mathbb{D}}\rightarrow{\mathbb{D}}_{\mathrm{wt}_{0}}\,.$

Virasoro constraints in sheaf theory have been studied in [BLM22] building on
the previous works [MOOP22, Mor22, vB23]. A proof of the Virasoro constraints
for the moduli spaces $M_{d,\chi}$ with $\gcd(d,\chi)=1$ is given in [LM24].

######  Theorem 3.1 ([LM24]).

Given $d$ and $\chi$ coprime, we have

$\int_{M_{d,\chi}}{\mathsf{L}}_{\mathrm{wt}_{0}}(D)=0\quad\text{for
all}\;\;D\in{\mathbb{D}}\,.$

For concrete calculations, it is useful to record here the normalized
formulation of the Virasoro constraints (cf. [BLM22, Conjecture 2.15]) in
terms of the generators $c_{k}(j)$. It has been explained in Section 1.2 how
the descendent algebra ${\mathbb{D}}_{\alpha}$ is identified with the
polynomial algebra ${\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]$.
In particular, the operators ${\mathsf{R}}_{n}$ and ${\mathsf{T}}_{n}$ can be
interpreted as operators acting on
${\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]$, and we have

${\mathsf{R}}_{n}(c_{k}(j))=(k+j-1)\cdots(k+j-1+n)c_{k+n}(j)\,.$

As discussed in Section 1.2, there is a normalized realization

${\mathbb{D}}_{\alpha}\simeq{\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]\to
H^{*}(M_{d,\chi})$

that sends $c_{1}(1)$ to zero. The normalized Virasoro operators
${\mathsf{L}}_{n}^{\delta}$ on
${\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]$ are given by

${\mathsf{L}}_{n}^{\delta}={\mathsf{R}}_{n}+{\mathsf{T}}_{n}-\frac{(n+1)!}{d}{\mathsf{R}}_{-1}\circ
c_{n+1}(1)\,.$

######  Corollary 3.2.

Given $d$ and $\chi$ coprime, we have

$\int_{M_{d,\chi}}{\mathsf{L}}_{n}^{\delta}(D)=0\quad\text{for all}\quad n\geq
0,\;\,D\in{\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]\,.$

###### Proof.

This follows from the standard form of the Virasoro constraints in Theorem 3.1
and [BLM22, Proposition 2.16]. Note that the definition of the realization of
$c_{k}(j)$ also involves the twisting operators ${\mathsf{F}}_{\rho}$ from
Section 1.2 and that these commute with the Virasoro operators by [BLM22,
Lemma 2.19]. ∎

###  3.2. Virasoro representation

Recall from the last section that we have a representation of the half of the
Virasoro algebra $\mathsf{Vir}_{\geq-1}$ on ${\mathbb{D}}$. On the other hand,
we have an exact sequence

$0\rightarrow{\mathfrak{I}}_{d,\chi}\rightarrow{\mathbb{D}}_{d,\chi}\rightarrow
H^{*}({\mathfrak{M}}_{d,\chi})\rightarrow 0\,.$

It turns out that as a consequence of Virasoro constraints, one can show that
the representation of $\mathsf{Vir}_{\geq-1}$ descends to the cohomology of
the stack.

######  Theorem 3.3 ([LM24]).

The representation of $\mathsf{Vir}_{\geq-1}$ on ${\mathbb{D}}$ preserves the
ideal of relations ${\mathfrak{I}}_{d,\chi}$. Hence, the operators
${\mathsf{R}}_{n}$ descend to the cohomology $H^{*}({\mathfrak{M}}_{d,\chi})$,
i.e. there is a dashed arrow completing the diagram

${{\mathbb{D}}}$${{\mathbb{D}}}$${H^{*}({\mathfrak{M}}_{d,\chi})}$${H^{*}({\mathfrak{M}}_{d,\chi}).}$$\scriptstyle{{\mathsf{R}}_{n}}$$\scriptstyle{\xi}$$\scriptstyle{\xi}$

######  Remark 3.4.

Note that $\mathsf{Vir}_{\geq-1}$ acts on ${\mathbb{D}}$ via either
$\\{{\mathsf{R}}_{n}\\}_{n\geq-1}$ or $\\{{\mathsf{L}}_{n}\\}_{n\geq-1}$.
Since the multiplication operator ${\mathsf{T}}_{n}$ trivially preserves the
ideal ${\mathfrak{I}}_{d,\chi}$, the above theorem is true for either choice
of a representation. In practice, however, we always work with the simpler one
using $\\{{\mathsf{R}}_{n}\\}_{n\geq-1}$.

######  Remark 3.5.

The most natural way to prove a statement like Theorem 3.3 is to utilize the
approximation of $H^{*}({\mathfrak{M}}_{d,\chi})$ by the cohomology of moduli
spaces of Joyce–Song pairs $P^{N}_{d,\chi}$, see the proof of Theorem 4.4. If
we knew the pair Virasoro constraints (cf. [BLM22, Conjecture 2.18]) for
Joyce–Song pairs, we could deduce that ${\mathsf{R}}_{n}$ descends to the
cohomology $H^{*}(P^{N}_{d,\chi})$, and hence to the cohomology of the stack
by approximation. The Virasoro constraints for $P^{N}_{d,\chi}$ are equivalent
to the Virasoro constraints for the Joyce invariant classes
$[M_{d,\chi}]^{\mathrm{inv}}$ for every $(d,\chi)$, but currently we only know
these unconditionally when $\gcd(d,\chi)=1$. This problem is worked around in
[LM24] by identifying ${\mathfrak{M}}_{d,\chi}$ with a stack of quiver
representations and approximating it by framed quiver representations, which
are the analogue of Joyce–Song pairs in the quiver context.

We can also make a statement for the cohomology of the good moduli spaces by
using the normalized realization. We have an exact sequence

$0\to I_{d,\chi}\to{\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]\to
H^{*}(M_{d,\chi})\to 0\,.$

Define the derivation ${\mathsf{R}}_{n}^{\delta}$ on
${\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]$ by

${\mathsf{R}}_{n}^{\delta}={\mathsf{R}}_{n}-\frac{(n+1)!}{d}c_{n+1}(1)\circ{\mathsf{R}}_{-1}\,.$

This derivation satisfies the property that
${\mathsf{R}}_{n}^{\delta}(c_{1}(1))=0$ for every $n\geq 0$, so it induces an
operator on ${\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]$ which we denote
in the same way. Note that
${\mathsf{L}}_{n}^{\delta}={\mathsf{R}}_{n}^{\delta}+{\mathsf{T}}_{n}^{\delta}$
where ${\mathsf{T}}_{n}^{\delta}={\mathsf{T}}_{n}-\frac{(n+1)!}{d}c_{n}(1)$ is
linear.

The following proposition is immediate:

######  Proposition 3.6.

Let
${\mathfrak{I}}\subseteq{\mathbb{D}}_{\alpha}\simeq{\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]$
be an ideal closed under the action of the operators ${\mathsf{R}}_{n}$,
$n\geq-1$. Then the ideal
$I\subseteq{\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]$ obtained as the
image of ${\mathfrak{I}}$ by the quotient setting $c_{1}(1)$ to 0 is closed
under the operators ${\mathsf{R}}_{n}^{\delta}$, $n\geq 0$.

###### Proof.

Since ${\mathsf{R}}_{n}$ and ${\mathsf{R}}_{-1}$ preserve the ideal
${\mathfrak{I}}$, the operator ${\mathsf{R}}_{n}^{\delta}$ also preserves the
ideal ${\mathfrak{I}}$, so the induced operator ${\mathsf{R}}_{n}^{\delta}$ on
the quotient ${\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]$ preserves the
ideal $I$. ∎

######  Corollary 3.7.

Let $d,\chi$ be coprime and $n\geq 0$. The derivation
${\mathsf{R}}_{n}^{\delta}$ preserves the ideal $I_{d,\chi}$.

###### Proof.

By Proposition 3.6 above, this is a consequence of the stacky analogue
(Theorem 3.3) and Lemma 1.16. However, it is instructive to give a proof
directly from the normalized form of the Virasoro constraints, which we do
below.

Suppose that $D\in I_{d,\chi}$ and let
$E\in{\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]$ be arbitrary. By the
Virasoro constraints in the form of Corollary 3.2,

$0=\int_{M_{d,\chi}}{\mathsf{L}}_{n}^{\delta}(D\cdot
E)=\int_{M_{d,\chi}}{\mathsf{R}}_{n}^{\delta}(D)\cdot
E+\int_{M_{d,\chi}}D\cdot{\mathsf{L}}_{n}^{\delta}(E)=\int_{M_{d,\chi}}{\mathsf{R}}_{n}^{\delta}(D)\cdot
E\,.$

Since $E$ was arbitrary and the realization map
${\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]\to H^{*}(M_{d,\chi})$ is
surjective, we conclude by Poincaré duality that
${\mathsf{R}}_{n}^{\delta}(D)\in I_{d,\chi}$. ∎

###  3.3. Preservation of geometric relations

In this subsection, we study preservation of the ideal
${\mathfrak{I}}^{\bullet}_{d,\chi}$ under the
$\mathsf{Vir}_{\geq-1}$-representation for each
$\bullet={\mathsf{MR}},{\mathsf{GMR}},{\mathsf{BR}},{\mathsf{Geom}}$. The
${\mathsf{BR}}$ case is trivial, and the ${\mathsf{Geom}}$ case follows from
the other three cases by definition. So we study the ${\mathsf{MR}}$ and
${\mathsf{GMR}}$ cases in the next two sections.

####  3.3.1. Mumford relations

We prove here the following proposition:

######  Proposition 3.8.

The ideal ${\mathfrak{I}}_{d,\chi}^{{\mathsf{MR}}}$ is closed under the action
of $U(\mathsf{Vir}_{\geq-1})$. More is true: for each choice of sign $\pm$ and
$k\in{\mathbb{Z}}$, the ideal generated by ${\mathsf{MR}}_{d,\chi}^{\pm,k,j}$
for $j>\pm\chi+kd$ is closed under the action of $U(\mathsf{Vir}_{\geq-1})$.

###### Proof.

The Virasoro operators are well behaved with respect to the isomorphisms
${\mathfrak{I}}_{d,\chi}\cong{\mathfrak{I}}_{d,\pm\chi+kd}$, so it is enough
to treat the case $\pm=+$ and $k=0$. To simplify notation, we write
$A_{j}={\mathsf{MR}}_{d,\chi}^{+,0,j}$. Recall that $A_{j}$ is the degree $2j$
term of

$A=\exp\left(\sum_{\ell\geq
1}(-1)^{\ell-1}(\ell-1)!{\mathrm{ch}}_{\ell}({\mathrm{td}}({\mathbb{P}}^{2}))\right)\,.$

To compute ${\mathsf{R}}_{n}(A_{j})$, we apply ${\mathsf{R}}_{n}$ to the
exponential:

${\mathsf{R}}_{n}(A)=\left(\sum_{\ell\geq
1}(-1)^{\ell-1}(\ell+n)!{\mathrm{ch}}_{\ell+n}({\mathrm{td}}({\mathbb{P}}^{2}))\right)A\,$

hence

${\mathsf{R}}_{n}(A_{j})=\sum_{\ell=1}^{j}(-1)^{\ell-1}(\ell+n)!{\mathrm{ch}}_{\ell+n}({\mathrm{td}}({\mathbb{P}}^{2}))A_{j-\ell}\,.$

On the other hand, we have also

$(n+j)A_{n+j}={\mathsf{R}}_{0}(A_{j+n})=\sum_{\ell=1}^{j+n}(-1)^{\ell-1}\ell!{\mathrm{ch}}_{\ell}({\mathrm{td}}({\mathbb{P}}^{2}))A_{j+n-\ell}\,.$

Assume first that $n\geq 0$. Comparing the two equations we find that

(15)
${\mathsf{R}}_{n}(A_{j})=(-1)^{n}(n+j)A_{n+j}+\sum_{\ell=1}^{n}(-1)^{n+\ell}\ell!{\mathrm{ch}}_{\ell}({\mathrm{td}}({\mathbb{P}}^{2}))A_{j+n-\ell}\,.$

If $j>\chi$ then $j+n-\ell\geq j>\chi$, so clearly the right hand side is also
in ${\mathfrak{I}}_{d,\chi}^{{\mathsf{MR}}}$. When $n=-1$, we obtain

${\mathsf{R}}_{-1}(A_{j})=\big{(}{\mathrm{ch}}_{0}({\mathrm{td}}({\mathbb{P}}^{2}))-j+1\big{)}A_{j-1}=\big{(}\chi-j+1\big{)}A_{j-1}\,.$

If $j>\chi+1$ then $j-1>\chi$ and
$A_{j-1}\in{\mathfrak{I}}_{d,\chi}^{{\mathsf{MR}}}$; when $j=\chi+1$ the
factor $\chi-j+1$ vanishes, so ${\mathsf{R}}_{-1}(A_{\chi+1})=0$ is also in
the ideal. ∎

By the definition of $I_{d,\chi}^{\mathsf{MR}}$ and Proposition 3.6 the
corollary below follows:

######  Corollary 3.9.

The ideal $I_{d,\chi}^{\mathsf{MR}}$ is closed under the action of the
operators ${\mathsf{R}}_{n}^{\delta}$ for $n\geq 0$.

####  3.3.2. Generalized Mumford relations

We proceed now to prove that the generalized Mumford relations are also
preserved by the action of the Virasoro operators.

######  Theorem 3.10.

Let $\alpha=(d,\chi)$, $\alpha^{\prime}=(d^{\prime},\chi^{\prime})$ satisfy
(13). Then the ideal
${\mathfrak{I}}_{\alpha}^{{\mathsf{GMR}},\alpha^{\prime}}$ is closed under the
action of $U(\mathsf{Vir}_{\geq-1})$.

###### Proof.

Recall that the ideal
${\mathfrak{I}}_{\alpha}^{{\mathsf{GMR}},\alpha^{\prime}}$ is generated by
relations of the form ${\mathsf{GMR}}^{\alpha^{\prime},j,A}_{\alpha}$ where
$j>dd^{\prime}$ and $A\in H_{*}({\mathfrak{M}}_{\alpha^{\prime}})$. They are
obtained by starting with the formal class
$C_{j}\in{\mathbb{D}}_{\alpha}\otimes{\mathbb{D}}_{\alpha^{\prime}}$,
realizing the second factor ${\mathbb{D}}_{\alpha^{\prime}}\to
H^{*}({\mathfrak{M}}_{\alpha}^{\prime})$ and then integrating against $A$. The
crucial ingredient in the proof is Theorem A.1, which we prove in the
appendix. This is a combinatorial identity which interacts $C$ with the action
of the Virasoro operators; it states that we have an equality

(16) $\displaystyle({\mathsf{R}}_{n}\otimes\textup{id})(C_{j})$
$\displaystyle+\sum_{k=-1}^{n}\binom{n+1}{k+1}(\textup{id}\otimes{\mathsf{R}}_{k})(C_{j+n-k})$
$\displaystyle=\sum_{0\leq a+b\leq
n}\sum_{i}(-1)^{d_{i}^{L}+1}\frac{a!(n-a)!}{(n-a-b)!}\big{(}{\mathrm{ch}}_{a}(\gamma_{i}^{L})\otimes{\mathrm{ch}}_{b}(\gamma_{i}^{R})\big{)}C_{j+n-a-b}$

in ${\mathbb{D}}_{\alpha}\otimes{\mathbb{D}}_{\alpha^{\prime}}$. Since the
operators ${\mathsf{R}}_{k}$ descend to
$H^{*}({\mathfrak{M}}_{\alpha^{\prime}})$ by Theorem 3.3, the equality (16)
can be also viewed in ${\mathbb{D}}_{\alpha}\otimes
H^{*}({\mathfrak{M}}_{\alpha^{\prime}})$. Since
$H_{*}({\mathfrak{M}}_{\alpha^{\prime}})$ is dual to
$H^{*}({\mathfrak{M}}_{\alpha^{\prime}})$, we have dual operators
${\mathsf{R}}_{n}^{\dagger}\colon H_{*}({\mathfrak{M}}_{\alpha^{\prime}})\to
H_{*}({\mathfrak{M}}_{\alpha^{\prime}})$ on homology. Applying
$\textup{id}\otimes\int_{A}$ to (16) we obtain

(17)
$\displaystyle{\mathsf{R}}_{n}({\mathsf{GMR}}^{\alpha^{\prime},j,A}_{\alpha})$
$\displaystyle+\sum_{k=-1}^{n}\binom{n+1}{k+1}{\mathsf{GMR}}^{\alpha^{\prime},j+n-k,{\mathsf{R}}_{k}^{\dagger}(A)}_{\alpha}=$
$\displaystyle\sum_{0\leq a+b\leq
n}\sum_{i}(-1)^{d_{i}^{L}+1}\frac{a!(n-a)!}{(n-a-b)!}{\mathrm{ch}}_{a}(\gamma_{i}^{L}){\mathsf{GMR}}_{\alpha}^{\alpha^{\prime}\\!,\,j+n-a-b,\,{\mathrm{ch}}_{b}(\gamma_{i}^{R})\cap
A}\,.$

Note that for $k\leq n$ and $a+b\leq n$ we have $j+n-k,\,j+n-a-b\geq
j>dd^{\prime}$, so we conclude that
${\mathsf{R}}_{n}({\mathsf{GMR}}^{\alpha^{\prime},j,A}_{\alpha})\in{\mathfrak{I}}_{\alpha}^{{\mathsf{GMR}},\alpha^{\prime}}$.∎

Once again, a similar statement can be made at the level of the good moduli
space:

######  Corollary 3.11.

The ideal $I_{\alpha}^{{\mathsf{GMR}},\alpha^{\prime}}$ is closed under the
action of the operators ${\mathsf{R}}_{n}^{\delta}$ for $n\geq 0$.

###  3.4. Primitive relations

Knowing that ${\mathfrak{I}}^{\bullet}_{\alpha}$ is closed under the action of
$\mathsf{Vir}_{\geq-1}$ can provide a strong simplification on the
computations. We now proceed to present and explain a considerably smaller set
of generators of the ideals ${\mathfrak{I}}_{\alpha}^{\bullet}$ when we regard
them as a ${\mathbb{D}}_{\alpha}\otimes U(\mathsf{Vir}_{\geq-1})$-module,
rather than just as an ideal in ${\mathbb{D}}_{\alpha}$. Here, the algebra
structure of ${\mathbb{D}}_{\alpha}\otimes U(\mathsf{Vir}_{\geq-1})$ is
induced from that of ${\mathbb{D}}_{\alpha}$ and $U(\mathsf{Vir}_{\geq-1})$
together with the commutation rule $[{\mathsf{R}}_{k},D]={\mathsf{R}}_{k}(D)$.

######  Definition 3.12.

We define the set $\mathfrak{P}^{\bullet}_{\alpha}$ of primitive relations of
three types as follows:

  1. (i)

$\mathfrak{P}_{\alpha}^{\mathsf{MR}}$ is the set of relations
${\mathsf{MR}}^{\pm,k,j}_{\alpha}$ with $\pm\chi+kd\geq g$ and
$j=\pm\chi+kd+1$.

  2. (ii)

$\mathfrak{P}_{\alpha}^{{\mathsf{GMR}},\alpha^{\prime}}$ is the set of
relations ${\mathsf{GMR}}^{\alpha^{\prime},j,A}_{\alpha}$ with either

$j=dd^{\prime}+1\textup{ or }\big{(}j=dd^{\prime}+2\textup{ and
}\deg(A)=dd^{\prime}+2\big{)}\,.$

  3. (iii)

$\mathfrak{P}_{\alpha}^{\mathsf{BR}}$ consists of the single relation
${\mathrm{ch}}_{1}(H^{2})^{b+1}$.

Clearly each of these sets are contained in the corresponding ideals including
non-primitive relations. For example, ${\mathfrak{I}}_{\alpha}^{\mathsf{MR}}$
is generated by all relations ${\mathsf{MR}}^{\pm,k,j}_{\alpha}$ with
$j\geq\pm\chi+kd+1$, but $\mathfrak{P}_{\alpha}^{\mathsf{MR}}$ consists only
of the first Chern class beyond the rank. It turns out that the primitive
relations are enough to generate all the relations once we know that the
ideals are closed under the Virasoro action.

######  Theorem 3.13.

The ideals
${\mathfrak{I}}_{\alpha}^{{\mathsf{MR}}},{\mathfrak{I}}_{\alpha}^{{\mathsf{GMR}},\alpha^{\prime}},{\mathfrak{I}}_{\alpha}^{{\mathsf{BR}}}$
are the smallest ideals containing $\mathfrak{P}_{\alpha}^{{\mathsf{MR}}}$,
$\mathfrak{P}_{\alpha}^{{\mathsf{GMR}},\alpha^{\prime}}$,
$\mathfrak{P}_{\alpha}^{{\mathsf{BR}}}$, respectively, that are closed under
the action of $U(\mathsf{Vir}_{\geq-1})$.

###### Proof.

We have already shown that the ideals
${\mathfrak{I}}_{\alpha}^{{\mathsf{MR}}}$,
${\mathfrak{I}}_{\alpha}^{{\mathsf{GMR}},\alpha^{\prime}}$,
${\mathfrak{I}}_{\alpha}^{{\mathsf{BR}}}$ are closed under the action of
$U(\mathsf{Vir}_{\geq-1})$ and they clearly contain the primitive relations,
so it is enough to show that they are the smallest such ones.

We start with the ideal of Mumford relations. As we did in the proof of
Proposition 3.8, it is enough to consider $k=0$ and $\pm=+$. Considering
equation (15) for $n=1$, it is clear that if $A_{j}$ is in some ideal closed
under ${\mathsf{R}}_{1}$, then $A_{j+1}$ is also in such ideal. Since
$A_{\chi+1}$ is by definition in $\mathfrak{P}_{\alpha}^{{\mathsf{MR}}}$, it
follows by induction on $j$ that $A_{j}$ is contained in the smallest ideal
containing $\mathfrak{P}_{\alpha}^{{\mathsf{MR}}}$ and closed under the
$U(\mathsf{Vir}_{\geq-1})$-action for every $j>\chi$.

Let us consider the case of generalized Mumford relations. First, we observe
that we can modify (16) by using Lemma A.2 to remove the $k=-1$ term, which
gives

(18)
${\mathsf{R}}_{n}({\mathsf{GMR}}^{\alpha^{\prime},j,A}_{\alpha})+\sum_{k=0}^{n}\binom{n+1}{k+1}{\mathsf{GMR}}^{\alpha^{\prime},j+n-k,{\mathsf{R}}_{k}^{\dagger}(A)}_{\alpha}=(j+n){\mathsf{GMR}}^{\alpha^{\prime},j+n,A}_{\alpha}+\cdots\,$

where $\cdots$ denotes the remaining terms on the right hand side of (17) with
$0<a+b\leq n$; note that all these terms are in the ideal generated by
relations of the form ${\mathsf{GMR}}^{\alpha^{\prime},i,A^{\prime}}_{\alpha}$
with $j\leq i<j+n$. The operator ${\mathsf{R}}_{0}$ is given by
${\mathsf{R}}_{0}(D)=\frac{1}{2}\deg(D)D$ where $\deg(D)\in{\mathbb{Z}}$ is
the cohomological degree, so
${\mathsf{R}}_{0}^{\dagger}(A)=\frac{1}{2}\deg(A)A$ where $\deg(A)$ is the
homological degree. Considering this, the “leading term”
${\mathsf{GMR}}_{\alpha}^{\alpha^{\prime},j+n,A}$ appears twice in (18): in
the $k=0$ term of the sum, and in the right hand side. If

(19) $\frac{1}{2}(n+1)\deg(A)\neq j+n$

then we can express ${\mathsf{GMR}}_{\alpha}^{\alpha^{\prime},j+n,A}$ in terms
of relations ${\mathsf{GMR}}_{\alpha}^{\alpha^{\prime},j^{\prime},A^{\prime}}$
with $j\leq j^{\prime}<j+n$. We now argue that every relation of the form
${\mathsf{GMR}}^{\alpha^{\prime},i,A}_{\alpha}$ is contained in any ideal
containing the primitive relations and closed under the Virasoro action. We
proceed by induction on $i$. If $i=dd^{\prime}+1$ then the relation
${\mathsf{GMR}}^{\alpha^{\prime},i,A}_{\alpha}$ is primitive, so there is
nothing to prove. If $i=dd^{\prime}+2$ then either $\deg(A)=dd^{\prime}+2$, in
which case ${\mathsf{GMR}}^{\alpha^{\prime},i,A}_{\alpha}$ is primitive by
definition, or $(j,n)=(i-1,1)$ satisfies the inequality (19), in which case we
conclude by induction. If $i\geq dd^{\prime}+3$ then the inequality (19) holds
for either $(j,n)=(i-1,1)$ or $(j,n)=(i-2,2)$, so we conclude again by
induction.

For base relations, we show by induction on $j\geq 0$ that any ideal
containing ${\mathrm{ch}}_{1}(H^{2})^{b+1}$ closed under the Virasoro action
contains every element of the form

(20)
${\mathrm{ch}}_{i_{1}}(H^{2})\cdots{\mathrm{ch}}_{i_{j}}(H^{2}){\mathrm{ch}}_{1}(H^{2})^{b+1-j}$

for any $i_{1},\ldots,i_{j}\geq 2$. Note that if we apply
${\mathsf{R}}_{i_{j+1}-1}$ to a term of the form (20) we obtain

$(b+1-j)i_{j+1}!{\mathrm{ch}}_{i_{1}}(H^{2})\cdots{\mathrm{ch}}_{i_{j}}(H^{2}){\mathrm{ch}}_{i_{j+1}}(H^{2}){\mathrm{ch}}_{1}(H^{2})^{b-j}+\Big{(}\textup{terms
of the form \eqref{eq: brinduction}}\Big{)}\,.$

This shows that the claim for $j$ implies the same claim for $j+1$, and
concludes the proof.∎

######  Remark 3.14.

The primitive generalized Mumford relations with $j=dd^{\prime}+2$ are indeed
necessary. For example, take $(d,\chi)=(2,1)$ and
$(d^{\prime},\chi^{\prime})=(1,0)$, the two (primitive) relations in degrees
$1$ and $2$ given by $j=dd^{\prime}+1=3$ are

(21) $c_{0}(2)-2c_{2}(0),\quad
2c_{2}(0)c_{1}(1)-2c_{2}(0)c_{0}(2)-c_{1}(1)c_{0}(2)+c_{0}(2)^{2}+2c_{1}(2)-4c_{3}(0).$

Applying $\mathsf{L}_{1}$ to $c_{0}(2)-2c_{2}(0)$, we obtain a second relation
$2c_{1}(2)-4c_{3}(0)$ in degree 2. Note that it lies in the ideal generated by
(21). On the other hand, the degree 2 relation obtained by taking
$j=4,\,\overline{A}=[M_{1,0}]$ is not in this ideal (by direct computation).
Thus we do need to include the latter into the primitive relations.

##  4\. BPS integrality

In this section, we recall cohomological Hall algebras (CoHA in short) and BPS
integrality of Mozgovoy–Reineke [MR15]. By establishing tautological
generation and purity of the cohomology of the moduli stacks, we relate the
Betti numbers of the moduli stacks to the intersection Betti numbers of the
good moduli spaces and vice versa. As applications, we prove structural
formulas for the Poincaré series of the moduli stacks and free generation of
cohomology rings in low degrees.

###  4.1. Cohomological Hall algebra

Denote the monoid of topological types of one-dimensional sheaves by

$\Lambda:=\\{(0,0)\\}\sqcup\,{\mathbb{Z}}_{\geq 1}\times{\mathbb{Z}}.$

For a fixed slope $\mu\in{\mathbb{Q}}$, we define the submonoid

$\Lambda_{\mu}:=\\{(d,\chi)\in\Lambda\,|\,\chi=d\cdot\mu\\}.$

Denote the set of nonzero elements by
$\Lambda_{\mu}^{\times}:=\Lambda_{\mu}-\\{(0,0)\\}$. Recall that semistable
one-dimensional sheaves of slope $\mu$ form an abelian category
$\mathcal{A}_{\mu}$. This is a hereditary category in the sense that
$\operatorname{Ext}^{2}(F_{1},F_{2})=0$ for any
$F_{1},F_{2}\in\mathcal{A}_{\mu}$ by semistability and Serre duality. Denote
the corresponding smooth moduli stack by

${\mathfrak{M}}_{\mu}:=\bigsqcup_{(d,\chi)\in\Lambda_{\mu}}{\mathfrak{M}}_{d,\chi}$

where ${\mathfrak{M}}_{0,0}$ is simply a point. Consider the stack of short
exact sequences in $\mathcal{A}_{\mu}$ and forgetful morphisms

(22)
${\text{SES}_{\mu}}$${{\mathfrak{M}}_{\mu}\times{\mathfrak{M}}_{\mu}}$${{\mathfrak{M}}_{\mu}.}$$\scriptstyle{\pi_{1}\times\pi_{3}}$$\scriptstyle{\pi_{2}}$

Precisely, the stack $\text{SES}_{\mu}$ parametrizes short exact sequences
$0\rightarrow F_{1}\rightarrow F_{2}\rightarrow F_{3}\rightarrow 0$ where all
$F_{i}$’s are semistable one-dimensional sheaves of slope $\mu$ and the
morphisms $\pi_{1}\times\pi_{3}$ and $\pi_{2}$ send this to $(F_{1},F_{3})$
and $F_{2}$, respectively. Since $\mathcal{A}_{\mu}$ is hereditary, we can use
diagram (22) to define the cohomological Hall algebra multiplication

$\star:=(\pi_{2})_{*}\circ(\pi_{1}\times\pi_{3})^{*}:H^{*}({\mathfrak{M}}_{\mu})\otimes
H^{*}({\mathfrak{M}}_{\mu})\rightarrow H^{*}({\mathfrak{M}}_{\mu}).$

We remark that the push-forward $(\pi_{2})_{*}$ is defined because $\pi_{2}$
is a proper morphism between smooth stacks. This indeed defines an associative
algebra with a unit $1\in{\mathbb{Q}}=H^{*}({\mathfrak{M}}_{0,0})\subset
H^{*}({\mathfrak{M}}_{\mu})$.

######  Definition 4.1.

We call $(H^{*}({\mathfrak{M}}_{\mu}),\star,1)$ the slope $\mu$ semistable
cohomological Hall algebra.

###  4.2. BPS integrality

One of the most important aspects of CoHA’s in various settings is the so-
called BPS integrality. Roughly speaking, BPS integrality relates the
cohomology of the moduli stacks and intersection cohomology of the moduli
spaces in the setting of hereditary categories. The precise meaning of BPS
integrality varies with the context. We use the BPS integrality theorem for
hereditary categories, following Mozgovoy–Reineke.

######  Theorem 4.2 ([MR15]).

We have an equality

(23)
$H^{*}({\mathfrak{M}}_{\mu},{\mathbb{Q}}^{\mathrm{vir}})={\textnormal{Sym}}\left(H^{*}({B}{\mathbb{G}}_{m},{\mathbb{Q}}^{\mathrm{vir}})\otimes\bigoplus_{(d,\chi)\in\Lambda_{\mu}^{\times}}H^{*}(M_{d,\chi},\mathrm{IC}^{\mathrm{vir}})\right)$

in $\widehat{K}(\mathrm{MHM}(\Lambda_{\mu}))$, the $\Lambda_{\mu}$-graded
localized $K$-theory of mixed Hodge modules.999The same equality holds in the
derived category of monodromic mixed Hodge modules
$D^{+}(\mathrm{MMHM}(\Lambda_{\mu}))$ if we consider semistable CoHA’s for
quivers without relations, see [DM20]. Equality in
$D^{+}(\mathrm{MMHM}(\Lambda_{\mu}))$ for global hereditary categories as in
our case seems to be unknown in the literature, even though this is expected
to hold.

In the rest of the subsection, we explain the terminology in this theorem. The
main language we use is Saito’s theory of mixed Hodge modules, see [Sai88] for
precise definitions. Let $X$ be a separated reduced scheme of finite type.
Then there is an abelian category of mixed Hodge modules $\mathrm{MHM}(X)$ and
its bounded derived category $D^{b}(\mathrm{MHM}(X))$. When $X={\mathsf{p}}$
is a point, $\mathrm{MHM}({\mathsf{p}})$ recovers the category of polarizable
mixed Hodge structures with rational coefficients. Important properties of
mixed Hodge modules include:

  1. (1)

There is a faithful functor $\mathbf{rat}:D^{b}(\mathrm{MHM}(X))\rightarrow
D^{b}_{\mathrm{cst}}(X)$ to the bounded derived category of constructible
sheaves.

  2. (2)

The functor $\mathbf{rat}$ sends $\mathrm{MHM}(X)$ to the category
$\mathrm{Perv}(X)$ of perverse sheaves.

  3. (3)

There exists a six functor formalism with Verdier duality $\mathbf{D}$ for
$D^{b}(\mathrm{MHM}(-))$.

  4. (4)

Every object $M\in\mathrm{MHM}(X)$ admits a weight filtration $W_{\bullet}M$.

We say that $M\in D^{b}(\mathrm{MHM}(X))$ is pure of weight $w$ if
$\mathrm{gr}^{W}_{i}{\mathcal{H}}^{j}(M)=0$ unless $i=j+w$. We simply say that
$M$ is pure if it is pure of weight $0$. The shift functor $[d]$ changes the
weight of a complex of mixed Hodge modules by $d$. The weight filtration
behaves nicely with respect to the six functors which in particular implies
the following.

  1. (5)

(Derived) push-forward along proper morphisms preserves pure complexes.

The following fact, called semisimplicity, is perhaps the deepest result about
mixed Hodge modules, which together with (5) recovers the decomposition
theorem [BBD82].

  1. (6)

If $M\in D^{b}(\mathrm{MHM}(X))$ is pure, then there is a (non-canonical)
isomorphism

$M\simeq\bigoplus_{j\in\mathbb{Z}}{\mathcal{H}}^{j}(M)[-j].$

Suppose moreover that $X$ is irreducible of dimension $d_{X}$. The
construction of the intersection complex as a perverse sheaf naturally lifts
to mixed Hodge modules. We also call such a lift the intersection complex and
denote it by $\mathrm{IC}_{X}\in\mathrm{MHM}(X)$. Consider the elements

${\mathbb{T}}\coloneqq{\mathbb{Q}}(-1)\in\mathrm{MHM}({\mathsf{p}}),\quad{\mathbb{L}}\coloneqq
H^{*}_{c}({\mathbb{A}}^{1})={\mathbb{Q}}(-1)[-2]\in
D^{b}(\mathrm{MHM}({\mathsf{p}}))$

which are used for the Tate twist and Lefschetz twist, respectively. We use
the same notations for the pull-backs of ${\mathbb{T}}$ and ${\mathbb{L}}$ via
$X\rightarrow{\mathsf{p}}$. When the dimension $d_{X}$ is even, the virtual
intersection complex is defined by

$\mathrm{IC}_{X}^{\mathrm{vir}}:={\mathbb{T}}^{-d_{X}/2}\otimes\mathrm{IC}_{X}\in\mathrm{MHM}(X).$

This complex is pure of weight $0$, so it is preserved by the Verdier duality
functor $\mathbf{D}$. If $X$ is smooth, the virtual intersection complex
becomes

$\mathrm{IC}_{X}^{\mathrm{vir}}={\mathbb{Q}}_{X}(d_{X}/2)[d_{X}]={\mathbb{L}}^{-d_{X}/2}.$

which we simply denote by ${\mathbb{Q}}^{\mathrm{vir}}_{X}$. When $d_{X}$ is
odd, we take the above definition valued in
$K(\mathrm{MHM}(X))[{\mathbb{L}}^{1/2}]$ where the square root
${\mathbb{L}}^{1/2}$, or equivalently ${\mathbb{T}}^{1/2}$, is formally
added.101010An alternative approach to this is to use the derived category of
monodromic mixed Hodge modules where the square root of the Tate twist exists,
see [Dav23, Section 2.1.6].

We now explain how the theory of mixed Hodge modules extends to stacks, as
required by Theorem 4.2. Since the abelian category of mixed Hodge modules
forms a sheaf with respect to the smooth topology [Ach13, Theorem 2.3], we can
define $\mathrm{MHM}({\mathfrak{X}})$ for a locally finite Artin stack
${\mathfrak{X}}$. The six functor formalism, however, is not defined in full
generality, hence requiring more care. This can be worked around for a certain
class of Artin stacks, including global quotient stacks by reductive groups,
see [Dav22] for details. According to such a definition, for example, we have

$H^{*}(B{\mathbb{G}}_{m},{\mathbb{Q}})=1\oplus{\mathbb{L}}\oplus{\mathbb{L}}^{2}+\cdots\in
D^{+}(\mathrm{MHM}({\mathsf{p}}))$

where $D^{+}(-)$ denotes the derived category bounded from below. The virtual
cohomology of $B{\mathbb{G}}_{m}$ defines an element

$H^{*}(B{\mathbb{G}}_{m},{\mathbb{Q}}^{\mathrm{vir}})=\frac{{\mathbb{L}}^{1/2}}{1-{\mathbb{L}}}\in\widehat{K}(\mathrm{MHM}({\mathsf{p}}))$

where $\widehat{K}(\mathrm{MHM}({\mathsf{p}}))$ denotes the localized
$K$-theory of mixed Hodge modules over a point defined by

(24) $\widehat{K}(\mathrm{MHM}({\mathsf{p}}))\coloneqq
K(\mathrm{MHM}({\mathsf{p}}))\otimes_{{\mathbb{Z}}[{\mathbb{L}}]}{\mathbb{Q}}[\,{\mathbb{L}}^{1/2},(1-{\mathbb{L}}^{n})^{-1}\,|\,n\geq
1].$

Finally, we explain the $\Lambda_{\mu}$-grading and symmetric product
${\textnormal{Sym}}(-)$ with respect to a certain symmetric monoidal
structure. Recall that $(\Lambda_{\mu},+)$ is a commutative monoid that is
abstractly isomorphic to ${\mathbb{Z}}_{\geq 0}$. We regard $\Lambda_{\mu}$ as
a commutative monoid object in the category of schemes by considering it as a
disjoint union of points, denoted by ${\mathsf{p}}_{\alpha}$ for each
$\alpha=(d,\chi)\in\Lambda_{\mu}$. Define the bounded below derived category
of $\Lambda_{\mu}$-graded mixed Hodge modules as

$D^{+}(\mathrm{MHM}(\Lambda_{\mu})):=\prod_{\alpha\in\Lambda_{\mu}}D^{+}(\mathrm{MHM}({\mathsf{p}}_{\alpha})).$

This is equipped with a monoidal structure given by

$D^{+}(\mathrm{MHM}({\mathsf{p}}_{\alpha_{1}}))\times
D^{+}(\mathrm{MHM}({\mathsf{p}}_{\alpha_{2}}))\rightarrow
D^{+}(\mathrm{MHM}({\mathsf{p}}_{\alpha_{1}+\alpha_{2}})),\quad(M_{1},M_{2})\mapsto+_{*}(M_{1}\boxtimes
M_{2})$

where
$+:{\mathsf{p}}_{\alpha_{1}}\times{\mathsf{p}}_{\alpha_{2}}\rightarrow{\mathsf{p}}_{\alpha_{1}+\alpha_{2}}$.
Moreover, this can be upgraded to a symmetric monoidal structure by [MSS11].
This induces a $\lambda$-ring structure and a symmetric product
${\textnormal{Sym}}(-)$ on $\widehat{K}(\mathrm{MHM}(\Lambda_{\mu}))$.

######  Remark 4.3.

The way we state Theorem 4.2 is not exactly the same as in the reference
[MR15]. Our formulation follows easily by applying the Verdier duality
$\mathbf{D}$ to Equation (12) combined with Theorem 5.4 of loc. cit. Note that
we need to use that the Verdier duality preserves the virtual intersection
complexes ${\mathbb{Q}}^{\mathrm{vir}}_{{\mathfrak{M}}_{d,\chi}}$ and
$\mathrm{IC}^{\mathrm{vir}}_{M_{d,\chi}}$. The authors illustrated their
theory in the case of semistable sheaves on curves, but it was explained that
the same result applies to any hereditary categories with an exact framing
functor such that the Euler form is symmetric, see [MR15, Remark 3.12]. These
assumptions are satisfied for the abelian category ${\mathcal{A}}_{\mu}$ of
semistable one-dimensional sheaves of slope $\mu$ on
${\mathbb{P}}^{2}$.111111The Euler form is symmetric for a trivial reason. We
also have a left exact framing functor
$\operatorname{Hom}({\mathcal{O}}(-N),-):{\mathcal{A}}_{\mu}\rightarrow\textup{Vect}$.
For any given $(d,\chi)\in\Lambda_{\mu}$, we can choose $N$ sufficiently large
so that this functor is exact up to degree $d$, and the proof of [MR15] can be
adapted to this setting. Alternatively, we can use the exact framing functor
$\operatorname{Ext}^{1}(F^{\prime},-):{\mathcal{A}}_{\mu}\rightarrow\textup{Vect}$
where $F^{\prime}$ is a fixed semistable sheaf such that
$\mu<\mu^{\prime}<\mu+3$.

###  4.3. Tautological generation

In this section, we prove the following properties of the cohomology of the
moduli stacks, including tautological generation, cf. Theorem 1.3 (ii).

######  Theorem 4.4.

The cohomology ring $H^{*}({\mathfrak{M}}_{d,\chi},{\mathbb{Q}})$ is
tautologically generated and pure. Furthermore, the cycle class map from Chow
to cohomology is an isomorphism.

To motivate our proof, we explain some of its main ingredients in the
analogous statement for the classifying stack
$B\mathrm{GL}_{n}\coloneqq[{\mathsf{p}}/\mathrm{GL}_{n}]$. Denote by
${\mathcal{V}}\rightarrow B\mathrm{GL}_{n}$ the universal rank $n$ vector
bundle. For each $N\geq n$, we have a diagram

(25)
${{\mathcal{H}}\kern-1.0pt\mathit{om}_{n}^{N}\coloneqq{\mathcal{H}}\kern-1.0pt\mathit{om}({\mathbb{C}}^{N}\otimes{\mathcal{O}}_{B\mathrm{GL}_{n}},{\mathcal{V}})}$${\mathrm{Gr}(N,n)}$${B\mathrm{GL}_{n}}$$\scriptstyle{\pi}$$\scriptstyle{j}$

where ${\mathcal{H}}\kern-1.0pt\mathit{om}_{n}^{N}$ is the total space of the
vector bundle over $B\mathrm{GL}_{n}$ and $\mathrm{Gr}(N,n)$ is the
Grassmannian. The stack ${\mathcal{H}}\kern-1.0pt\mathit{om}_{n}^{N}$ can be
interpreted as a stack parametrizing all morphisms
${\mathbb{C}}^{N}\rightarrow V$ with $\dim(V)=n$. Then
$j\colon\mathrm{Gr}(N,n)\rightarrow{\mathcal{H}}\kern-1.0pt\mathit{om}_{n}^{N}$
is an open embedding that corresponds to surjective morphisms
${\mathbb{C}}^{N}\twoheadrightarrow V$. The complement of $j$ has codimension
at most

$Nn-(N(n-1)+n-1)=N-n+1$

since it parametrizes morphisms ${\mathbb{C}}^{N}\rightarrow V$ that factor
through some $V^{\prime}\subsetneq V$. The geometry of the above diagram thus
implies

$\mathrm{CH}^{i}(B\mathrm{GL}_{n})\xlongrightarrow[\,\raisebox{1.99997pt}{\smash{$\sim$}}\,]{\pi^{*}}\mathrm{CH}^{i}({\mathcal{H}}\kern-1.0pt\mathit{om}_{n}^{N})\xlongrightarrow[\,\raisebox{1.99997pt}{\smash{$\sim$}}\,]{j^{*}}\mathrm{CH}^{i}(\mathrm{Gr}(N,n)),\quad
N-n+1>i$

where the isomorphisms come from homotopy invariance and excision,
respectively. On the other hand, a similar argument gives an isomrophism
$H^{\leq 2i}(B\mathrm{GL}_{n})\simeq H^{\leq 2i}(\mathrm{Gr}(N,n))$ for
$N-n+1>i$. Consequently, the cohomology and Chow groups of $B\mathrm{GL}_{n}$
can be understood via those of $\mathrm{Gr}(N,n)$ if $N$ is sufficiently
large. This reduces problems about tautological generation, purity and the
cycle class map for $B\mathrm{GL}_{n}$ to those of $\mathrm{Gr}(N,n)$, which
are better understood.

We follow the same strategy to study ${\mathfrak{M}}_{d,\chi}$. We first
introduce the moduli of limit stable pairs, also known as Joyce–Song pairs,
that plays the role of Grassmannians above. Let $N$ be an integer such that

(26) $dN+\chi\geq g=\frac{(d-1)(d-2)}{2}.$

Then Proposition 2.2 implies

(27)
$\operatorname{Ext}^{1}({\mathcal{O}}(-N),F)=\operatorname{Ext}^{2}({\mathcal{O}}(-N),F)=0$

for any semistable sheaf $F$ of type $(d,\chi)$.

######  Definition 4.5.

Assume that $N$ and $(d,\chi)$ satisfy (26). We say that a pair $(F,\phi)$ of
a one-dimensional sheaf $F$ of type $(d,\chi)$ and a nonzero map
$\phi:{\mathcal{O}}(-N)\rightarrow F$ is limit stable if

  1. (i)

$F$ is semistable, and

  2. (ii)

$\phi$ does not factor through any destabilizing subsheaf of $F$.

A morphism between two of such pairs $(F,\phi)$ and
$(F^{\prime},\phi^{\prime})$ is a morphism $f:F\rightarrow F^{\prime}$ such
that $f\circ\phi=\phi^{\prime}$.

######  Remark 4.6.

A morphism $f:F\rightarrow F^{\prime}$ between two limit stable pairs
$(F,\phi)$ and $(F^{\prime},\phi^{\prime})$ of identical topological type is
necessarily an isomorphism. This is because if $f$ is not an isomorphism nor
zero, then the image of $f$ defines a destabilizing subsheaf of $F^{\prime}$
that $\phi^{\prime}$ factors through.

The limit stable pairs form a projective moduli space, denoted by
$P^{N}_{d,\chi}$, whose construction via geometric invariant theory is due to
[LP93b]. See also [Lin18] for a modern account in a generalized setting.

###### Proof of Theorem 4.4.

Assume throughout the proof that $N$ and $(d,\chi)$ satisfy (26). We first
construct a diagram analogous to (25). By the vanishing (27), we obtain a
vector bundle

${\mathcal{H}}\kern-1.0pt\mathit{om}^{N}_{d,\chi}\coloneqq
p_{*}{\mathcal{H}}\kern-1.0pt\mathit{om}(q^{*}{\mathcal{O}}(-N),{\mathcal{F}})\overset{\pi}{\longrightarrow}{\mathfrak{M}}_{d,\chi}.$

The total space ${\mathcal{H}}\kern-1.0pt\mathit{om}^{N}_{d,\chi}$ of this
vector bundle is a stack parametrizing pairs of a semistable sheaf $F$ of type
$(d,\chi)$ together with a morphism $\phi:{\mathcal{O}}(-N)\rightarrow F$.
Since limit stable pairs form an open subset among all such pairs, we obtain a
diagram

(28)
${{\mathcal{H}}\kern-1.0pt\mathit{om}_{d,\chi}^{N}}$${P^{N}_{d,\chi}}$${{\mathfrak{M}}_{d,\chi}}$$\scriptstyle{\pi}$$\scriptstyle{j}$

where $\pi$ is a vector bundle and $j$ in an open embedding. We note that
$P^{N}_{d,\chi}$ is smooth because it is an open subset inside the total space
of a vector bundle over a smooth stack.

We now estimate the codimension of the complement of $j$. For each semistable
sheaf $F$ of type $(d,\chi)$, we need to study the closed inclusion of the
unstable locus

(29)
$\operatorname{Hom}({\mathcal{O}}(-N),F)^{\textnormal{unstable}}\hookrightarrow\operatorname{Hom}({\mathcal{O}}(-N),F),$

i.e., the locus of morphisms that factor through some destabilizing subsheaf
$F^{\prime}\subsetneq F$. Note that the dimension of the space of
destabilizing subsheaves $F^{\prime}\subsetneq F$ can be bounded from above by
a constant $c=c(d,\chi)$ that only depends on $(d,\chi)$. For example, such an
upper bound can be obtained by the maximum dimension of Quot schemes
$\mathrm{Quot}_{{\mathbb{P}}^{2}}(F,(d^{\prime},\chi^{\prime}))$ for each $F$
and $1\leq d^{\prime}<d$ with $\chi/d=\chi^{\prime}/d^{\prime}$. The precise
bound will not be relevant to the proof. On the other hand, for each
destabilizing subsheaf $F^{\prime}\subsetneq F$ of type
$(d^{\prime},\chi^{\prime})$ we have

$\displaystyle\dim\operatorname{Hom}({\mathcal{O}}(-N),F^{\prime})$
$\displaystyle=\chi({\mathcal{O}}(-N),F^{\prime})$
$\displaystyle=\frac{d^{\prime}}{d}(dN+\chi)$
$\displaystyle\leq\frac{d-1}{d}(dN+\chi).$

Combining these estimates, we obtain that

$\dim\operatorname{Hom}({\mathcal{O}}(-N),F)-\dim\operatorname{Hom}({\mathcal{O}}(-N),F)^{\textnormal{unstable}}\\\
\geq(dN+\chi)-\left(c(d,\chi)+\frac{d-1}{d}(dN+\chi)\right)\eqqcolon
N-\tilde{c}(d,\chi).$

In other words, the codimension of the complement of $j$ is bounded below by
$N-\tilde{c}(d,\chi)$ where $\tilde{c}$ depends only on $(d,\chi)$. In
particular, it grows arbitrarily large as $N$ does.

By homotopy invariance and the excision sequence again, as in the case of
$B\mathrm{GL}_{n}$, we conclude that the diagram (28) implies

(30) $\mathrm{CH}^{\leq i}({\mathfrak{M}}_{d,\chi})\simeq\mathrm{CH}^{\leq
i}(P^{N}_{d,\chi})\,,\quad H^{\leq 2i}({\mathfrak{M}}_{d,\chi})\simeq H^{\leq
2i}(P^{N}_{d,\chi}),\quad\textnormal{if}\ \ N-\tilde{c}(d,\chi)>i.$

For any given $i\geq 0$, we can find a sufficiently large $N$ that realizes
the above isomorphisms. So it suffices to prove the statement of the theorem
for the moduli space of limit stable pairs $P^{N}_{d,\chi}$. This is done in
the next proposition, finishing the proof. ∎

######  Proposition 4.7.

Assume that $N$ and $(d,\chi)$ satisfy (26). The cohomology ring of
$P^{N}_{d,\chi}$ is tautologically generated and pure. Furthermore, the cycle
class map from Chow to cohomology is an isomorphism.

###### Proof.

Purity of the cohomology follows directly from smoothness and projectivity of
$P=P^{N}_{d,\chi}$. Note that there exists a universal pair

$\Phi:q^{*}{\mathcal{O}}(-N)\rightarrow{\mathcal{F}}$

over $P\times{\mathbb{P}}^{2}$ by restricting that from
${\mathcal{H}}\kern-1.0pt\mathit{om}^{N}_{d,\chi}\times{\mathbb{P}}^{2}$. By
tautological classes of $P$ in either Chow or cohomology, we mean those
classes obtained from ${\mathcal{F}}$ just like in the case of moduli stack of
sheaves. Since ${\mathcal{F}}$ is indeed a pull-back from
${\mathfrak{M}}_{d,\chi}\times{\mathbb{P}}^{2}$, the notion of tautological
classes for ${\mathfrak{M}}_{d,\chi}$ and $P^{N}_{d,\chi}$ agrees in both Chow
and cohomology in the range (30) where they are isomorphic.

To prove statements about tautological generation and the cycle class map, we
shall construct an explicit Chow–Künneth decomposition of the diagonal

$\Delta\in\mathrm{CH}^{k}(P_{1}\times P_{2}),\quad k:=\dim(P).$

The subscripts in $P_{1}$ and $P_{2}$ are just to distinguish the two copies
of $P$. Denote the perfect complex coming from the universal pair by

${\mathcal{S}}^{\bullet}\coloneqq\Big{[}q^{*}{\mathcal{O}}(-N)\xrightarrow{\Phi}{\mathcal{F}}\Big{]}\in
D^{b}(P\times{\mathbb{P}}^{2})$

which is placed in degrees $[0,1]$. We consider the complex

$R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{S}}^{\bullet}_{1},{\mathcal{F}}_{2})\in
D^{b}(P_{1}\times P_{2})$

where $p:P_{1}\times P_{2}\times{\mathbb{P}}^{2}\rightarrow P_{1}\times P_{2}$
is the projection map and ${\mathcal{S}}^{\bullet}_{i}$ and
${\mathcal{F}}_{i}$ are pull-backs from $P_{i}\times{\mathbb{P}}^{2}$. We
claim that
$R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{S}}^{\bullet}_{1},{\mathcal{F}}_{2})$
is a vector bundle of rank $k$ admitting a tautological section $\tau$ that
cuts out the diagonal $P=\mathrm{Zero}(\tau)\hookrightarrow P\times P$.
Assuming the claim for the moment, we obtain the formula

$\displaystyle\Delta$
$\displaystyle=c_{k}(R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{S}}^{\bullet}_{1},{\mathcal{F}}_{2}))$
$\displaystyle=\left[\exp\left(\sum_{i\geq
1}(-1)^{i-1}(i-1)!\sum_{a+b=i}{\mathrm{ch}}^{({\mathcal{S}}^{\bullet}_{1})^{\vee}}_{a}{\mathrm{ch}}^{{\mathcal{F}}_{2}}_{b}({\mathrm{td}}({\mathbb{P}}^{2}))\right)\right]_{k}\,.$

Here
${\mathrm{ch}}^{({\mathcal{S}}^{\bullet}_{1})^{\vee}}_{a}{\mathrm{ch}}^{{\mathcal{F}}_{2}}_{b}({\mathrm{td}}({\mathbb{P}}^{2}))$
denotes121212For a $K$-theory class ${\mathcal{V}}\in
K^{0}(P\times{\mathbb{P}}^{2})$, the notation $\xi_{{\mathcal{V}}}(-)$ means
that we use ${\mathrm{ch}}({\mathcal{V}})$ for the realization.

$\sum_{i}\xi_{({\mathcal{S}}^{\bullet}_{1})^{\vee}}({\mathrm{ch}}_{a}(\gamma_{i}^{L}))\xi_{{\mathcal{F}}_{2}}({\mathrm{ch}}_{b}(\gamma_{i}^{R}))\in
H^{\ast}(P)\,,$

where
$\Delta_{\ast}{\mathrm{td}}({\mathbb{P}}^{2})=\sum_{i}\gamma_{i}^{L}\otimes\gamma_{i}^{R}$,
see Lemma 2.7 for a similar statement. This yields an explicit Chow–Künneth
decomposition of the diagonal in the tautological classes after expanding the
formula. Note that this argument uses the Chow–Künneth decomposition of the
diagonal of ${\mathbb{P}}^{2}$. By standard arguments, see for example
[Bea95], this implies tautological generation and the cycle class map being an
isomorphism for $P$.

We finish the proof by establishing the above claim. To show the vector bundle
property, it suffices to prove

(31)
$\operatorname{Ext}^{i}\big{(}[{\mathcal{O}}(-N)\xrightarrow{\phi_{1}}F_{1}]\,,\,F_{2}\big{)}=0$

for all $i\neq 0$ and $(F_{j},\phi_{j})\in P$ with $j=1,2$. This can be proved
by using the long exact sequence obtained by applying
$\operatorname{Hom}(-,F_{2})$ to the exact triangle

$S^{\bullet}_{1}\xrightarrow{\ i_{1}\ }{\mathcal{O}}(-N)\xrightarrow{\
\phi_{1}\ }F_{1}\xrightarrow{\ +1\ }.$

Both (i) and (ii) in Definition 4.5 are used to show the vanishing. The
argument is standard, see for example [BLM22, Lemma 5.3]. The only difference
is that we need to use an extra vanishing
$\operatorname{Ext}^{2}(F_{1},F_{2})=0$. The rank of the vector bundle can be
computed by $\chi(S^{\bullet},F)=\dim(P)$.

Over $P_{1}\times P_{2}\times{\mathbb{P}}^{2}$, consider the following
morphism given by the composition

$\tilde{\tau}:{\mathcal{S}}^{\bullet}_{1}\xrightarrow{\ I_{1}\
}{\mathcal{O}}(-N)\xrightarrow{\ \Phi_{2}\ }{\mathcal{F}}_{2}.$

Via adjunctions, this induces a tautological section
${\mathcal{O}}_{P_{1}\times
P_{2}}\xrightarrow{\tau}R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{S}}^{\bullet}_{1},{\mathcal{F}}_{2})$
of the vector bundle. We show that $Z\coloneqq\mathrm{Zero}(\tau)$ equals the
diagonal $\Delta$ scheme-theoretically. Since
$\tilde{\tau}\big{|}_{\Delta\times{\mathbb{P}}^{2}}=\Phi\circ I=0$, we have
$\Delta\hookrightarrow Z$. On the other hand, we have the following diagram

(32)
${{\mathcal{S}}^{\bullet}_{1}\big{|}_{Z\times{\mathbb{P}}^{2}}}$${{\mathcal{O}}(-N)}$${{\mathcal{F}}_{1}\big{|}_{Z\times{\mathbb{P}}^{2}}}$${{\mathcal{S}}^{\bullet}_{2}\big{|}_{Z\times{\mathbb{P}}^{2}}}$${{\mathcal{O}}(-N)}$${{\mathcal{F}}_{2}\big{|}_{Z\times{\mathbb{P}}^{2}}}$$\scriptstyle{I_{1}}$$\scriptstyle{\Phi_{1}}$$\scriptstyle{+1}$$\scriptstyle{I_{2}}$$\scriptstyle{\Phi_{2}}$$\scriptstyle{+1}$

over $Z\times{\mathbb{P}}^{2}$. By definition of $Z$, we have $\Phi_{2}\circ
I_{1}=0$ where we suppressed the restriction to $Z\times{\mathbb{P}}^{2}$ from
the notation. Since
$R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{S}}^{\bullet}_{1},{\mathcal{F}}_{2})$
is a vector bundle, local-to-global spectral sequence implies that

$\operatorname{Ext}^{-1}_{Z\times{\mathbb{P}}^{2}}({\mathcal{S}}^{\bullet}_{1}\big{|}_{Z\times{\mathbb{P}}^{2}},{\mathcal{F}}_{2}\big{|}_{Z\times{\mathbb{P}}^{2}})=H^{-1}(R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{S}}^{\bullet}_{1},{\mathcal{F}}_{2})\big{|}_{Z\times{\mathbb{P}}^{2}})=0.$

Therefore, we have a part of the long exact sequence

$0\rightarrow\operatorname{Hom}({\mathcal{S}}^{\bullet}_{1}\big{|}_{Z\times{\mathbb{P}}^{2}},{\mathcal{S}}^{\bullet}_{2}\big{|}_{Z\times{\mathbb{P}}^{2}})\xrightarrow{I_{2*}}\operatorname{Hom}({\mathcal{S}}^{\bullet}_{1}\big{|}_{Z\times{\mathbb{P}}^{2}},{\mathcal{O}}(-N))\xrightarrow{\Phi_{2*}}\operatorname{Hom}({\mathcal{S}}^{\bullet}_{1}\big{|}_{Z\times{\mathbb{P}}^{2}},{\mathcal{F}}_{2}\big{|}_{Z\times{\mathbb{P}}^{2}})\rightarrow\cdots$

where we used the above vanishing for the leftmost zero. Since $I_{1}$ in the
middle term is mapped to $\Phi_{2}\circ I_{1}=0$, there exists unique
$g\in\operatorname{Hom}({\mathcal{S}}^{\bullet}_{1}\big{|}_{Z\times{\mathbb{P}}^{2}},{\mathcal{S}}^{\bullet}_{2}\big{|}_{Z\times{\mathbb{P}}^{2}})$
such that $I_{2}\circ g=I_{1}$. In other words, the vertical map $g$ makes the
left square in (32) commutes. By the axioms of triangulated category, this
induces a morphism
$h\in\operatorname{Hom}({\mathcal{F}}^{\bullet}_{1}\big{|}_{Z\times{\mathbb{P}}^{2}},{\mathcal{F}}^{\bullet}_{2}\big{|}_{Z\times{\mathbb{P}}^{2}})$
making the right square commutes in (32). Such a morphism $h$ is necessarily
an isomorphism by the relative version of Remark 4.6. The universal property
of the moduli space $P=P^{N}_{d,\chi}$ then implies that a family of limit
stable pairs
${\mathcal{O}}(-N)\xrightarrow{\Phi_{i}}{\mathcal{F}}_{i}\big{|}_{Z\times{\mathbb{P}}^{2}}$
for $i=1,2$ defines the same morphism $Z\rightarrow P_{i}=P$. In other words,
$Z\rightarrow P_{1}\times P_{2}$ factors through the diagonal $\Delta$,
completing the proof. ∎

######  Remark 4.8.

The results in this section, i.e. Theorem 4.4 and Proposition 4.7, apply to
more general settings. For example, let $S$ be a smooth del Pezzo surface, $H$
an ample divisor and $\alpha$ a topological type of either positive rank or
one-dimensional sheaves. Then $\operatorname{Ext}^{2}_{S}(F_{1},F_{2})=0$ for
all $F_{1},F_{2}\in{\mathfrak{M}}_{\alpha}^{H{\textup{-ss}}}(S)$, which in
particular implies that ${\mathfrak{M}}_{\alpha}^{H{\textup{-ss}}}(S)$ is
smooth. By picking a positive enough line bundle $L$, we can construct the
limit stable pair moduli space $P_{\alpha}^{L}(S)$ which is smooth projective.
Then the same proof of Proposition 4.7 applies to show that
$H^{*}(P^{L}_{\alpha}(S))$ is pure, tautologically generated and isomorphic to
the Chow ring. The same properties hold for the moduli stack
${\mathfrak{M}}_{\alpha}^{H\textup{-ss}}(S)$ because we can identify
$H^{i}({\mathfrak{M}}_{\alpha}^{H\textup{-ss}}(S))\simeq
H^{i}(P_{\alpha}^{L}(S))$ where $L$ is chosen sufficiently positive according
to the cohomological degree $i$.

###  4.4. Applications

In this subsection, we apply results from Sections 4.2 and 4.3 to the
structure of the cohomology ring of the moduli stacks. See [MR15] for the
related discussion.

We define the virtual Hodge polynomial of $M\in\mathrm{MHM}({\mathsf{p}})$,
viewed as a mixed Hodge structure, by

$E(M)=E(M,u,v):=\sum_{p,q\in{\mathbb{Z}}}\dim
h^{p,q}(\mathrm{gr}^{W}_{p+q}M)u^{p}v^{q}\in{\mathbb{Z}}[u^{\pm},v^{\pm}].$

The virtual Hodge polynomial satisfies the short exact sequence relation,
hence extends to the $K$-theory $K(\mathrm{MHM}({\mathsf{p}}))$. In fact, this
defines a ring homomorphism

$E(-):K(\mathrm{MHM}({\mathsf{p}}))\rightarrow{\mathbb{Z}}[u^{\pm},v^{\pm}].$

We further extend this homomorphism to the localized $K$-theory (24)

$E(-):\widehat{K}(\mathrm{MHM}({\mathsf{p}}))\rightarrow{\mathbb{Z}}[u^{\pm},v^{\pm},(uv)^{1/2},(1-(uv)^{n})^{-1}\mid
n\geq 1]$

by letting

$E({\mathbb{L}}^{1/2},u,v)=-(uv)^{1/2},\quad
E((1-{\mathbb{L}}^{n})^{-1},u,v)=(1-(uv)^{n})^{-1}.$

Denote by ${\mathbb{Q}}\llbracket\Lambda_{\mu}\rrbracket$ the (completed)
monoid ring associated to $\Lambda_{\mu}$ whose elements are of the form
$\sum_{\alpha\in\Lambda_{\mu}}c_{\alpha}\cdot e^{\alpha}$ for a collection of
$c_{\alpha}\in{\mathbb{Q}}$.131313We consider the completion to be compatible
with our definition of $\widehat{K}(\mathrm{MHM}(\Lambda_{\mu}))$, which is
defined as a product
$\prod_{\lambda}\widehat{K}(\mathrm{MHM}({\mathsf{p}}_{\lambda}))$ rather than
a direct sum.  The Hodge polynomial homomorphism naturally extends to the
$\Lambda_{\mu}$-graded version by

$E(-):\widehat{K}(\mathrm{MHM}(\Lambda_{\mu}))\rightarrow{\mathbb{Q}}\llbracket\Lambda_{\mu}\rrbracket\otimes{\mathbb{Z}}[u^{\pm},v^{\pm},(uv)^{1/2},(1-(uv)^{n})^{-1}\mid
n\geq 1].$

Denote the shifted (intersection) Hodge polynomial of the moduli stacks and
spaces by