$\overline{E}({\mathfrak{M}}_{d,\chi}):=E(H^{*}({\mathfrak{M}}_{d,\chi},{\mathbb{Q}}^{\mathrm{vir}}),u,v),\quad\overline{\mathit{\mathit{IE}}}(M_{d,\chi}):=E(H^{*}(M_{d,\chi},\mathrm{IC}^{\mathrm{vir}}),u,v).$

Note that we have

$\overline{E}(B{\mathbb{G}}_{m})=\frac{-(uv)^{1/2}}{1-uv}.$

By applying the homomorphism $E(-)$ to the BPS integraity (23), we obtain the
following

######  Corollary 4.9.

We have an equality

(33)
$\sum_{(d,\chi)\in\Lambda_{\mu}}\overline{E}({\mathfrak{M}}_{d,\chi},u,v)\cdot
e^{(d,\chi)}=\mathrm{PE}\left(\frac{-(uv)^{1/2}}{1-uv}\cdot\sum_{(d,\chi)\in\Lambda_{\mu}^{\times}}\overline{\mathit{\mathit{IE}}}(M_{d,\chi},u,v)\cdot
e^{(d,\chi)}\right)$

in
${\mathbb{Q}}\llbracket\Lambda_{\mu}\rrbracket\otimes{\mathbb{Z}}[u^{\pm},v^{\pm},(uv)^{1/2},(1-(uv)^{n})^{-1}\mid
n\geq 1]$.

Here, the notation $\mathrm{PE}(-)$ denotes the plethystic exponential for the
variables $u,v$ and $e^{\alpha}$. Explicitly, it is given by

$\mathrm{PE}(f(u,v,e^{\alpha})):=\exp\left(\sum_{n\geq
1}\frac{f(u^{n},v^{n},e^{n\alpha})}{n}\right)$

where
$f\in{\mathbb{Q}}\llbracket\Lambda_{\mu}\rrbracket\otimes{\mathbb{Z}}[u^{\pm},v^{\pm},(uv)^{1/2},(1-(uv)^{n})^{-1}\,|\,n\geq
1]$ has zero coefficient for the $e^{0}$ component.

######  Remark 4.10.

Since $E(-,u,v)$ is defined using the graded factors with respect to the
weight filtration, it does not necessarily remember the cohomologically graded
dimensions. In the case of pure Hodge structures, however, taking graded
factors does not lose anything, so $E(-,u,v)$ can recover the information on
cohomological gradings. Note that the purity of
$H^{i}({\mathfrak{M}}_{d,\chi})$ is proved in Theorem 4.4 and the intersection
cohomology of a proper variety is always pure. Therefore, Corollary 4.9
relates two sets of numerical data

Betti numbers of $\\{{\mathfrak{M}}_{d,\chi}\\}_{(d,\chi)\in\Lambda_{\mu}}$
$\longleftrightarrow$ intersection Betti numbers of
$\\{M_{d,\chi}\\}_{(d,\chi)\in\Lambda_{\mu}^{\times}}$.

In fact, both sides of the formula in (33) depend only on the product $uv$
rather than the individual $u$ and $v$, because the Hodge structures involved
are not just pure but also algebraic, in the sense that they are of
$(p,p)$-type. This was proved in Theorem 4.4 for
$H^{*}({\mathfrak{M}}_{d,\chi})$, and the case for the intersection cohomology
of the moduli spaces follows from [Bou22, Theorem 0.4.1]. Therefore, we do not
lose any information by substituting $u=v=q^{1/2}$ and $(uv)^{1/2}=q^{1/2}$ to
(33), which reads

(34)
$\sum_{(d,\chi)\in\Lambda_{\mu}}\overline{E}({\mathfrak{M}}_{d,\chi},q)\cdot
e^{(d,\chi)}=\mathrm{PE}\left(\frac{-q^{1/2}}{1-q}\cdot\sum_{(d,\chi)\in\Lambda_{\mu}^{\times}}\overline{\mathit{\mathit{IE}}}(M_{d,\chi},q)\cdot
e^{(d,\chi)}\right)$

in
${\mathbb{Q}}\llbracket\Lambda_{\mu}\rrbracket\otimes{\mathbb{Z}}[q^{\pm},q^{1/2},(1-q^{n})^{-1}\mid
n\geq 1]$. Note that $\overline{E}(-,q)$ and
$\overline{\mathit{\mathit{IE}}}(-,q)$ are simply the shifted (intersection)
Poincaré series

$\displaystyle\overline{E}({\mathfrak{M}}_{d,\chi},q)$
$\displaystyle=(-q^{1/2})^{-\dim{\mathfrak{M}}_{d,\chi}}\sum_{i\geq 0}\dim
H^{2i}({\mathfrak{M}}_{d,\chi})\cdot q^{i},$
$\displaystyle\overline{\mathit{\mathit{IE}}}(M_{d,\chi},q)$
$\displaystyle=(-q^{1/2})^{-\dim M_{d,\chi}}\sum_{i\geq
0}\dim\mathit{IH}^{2i}(M_{d,\chi})\cdot q^{i}$

by the purity and algebraicity of the (intersection) cohomology. We will use
the notation $E({\mathfrak{M}}_{d,\chi},q)$ and $\mathit{IE}(M_{d,\chi},q)$
for the non-shifted Poincaré series, i.e. without the $(-q^{1/2})^{-\dim}$
factors.

The BPS integrality formula (34) (combined with Theorem 0.9) allows us to
compute Betti numbers of the stacks using those of the coprime moduli spaces,
as explained in Section 0.4. This gives a key completeness criterion for
geometric relations in our computation of their cohomology rings. Furthermore,
the BPS integrality formula implies strong structural results on the
cohomology of ${\mathfrak{M}}_{d,\chi}$ as we explain next.

######  Theorem 4.11.

The (non-shifted) Poincaré series $E({\mathfrak{M}}_{d,\chi},q)$ depends on
$(d,\chi)$ only through its degree $d$ and its multiplicity
$m\coloneqq\gcd(d,\chi)$. More precisely, there exists a polynomial

$A_{d,m}(q)=1+\cdots+(-1)^{m-1}q^{N},\quad N=d^{2}+\frac{m(m+1)}{2}$

such that

$E({\mathfrak{M}}_{d,\chi},q)=\frac{A_{d,m}(q)}{(1-q)(1-q^{2})\cdots(1-q^{m})}.$

###### Proof.

The first statement, concerning the dependence of
$E({\mathfrak{M}}_{d,\chi},q)$, follows immediately from the BPS integrality
formula (34) and the $\chi$-independence theorem for intersection cohomology
[MS23, Theorem 0.1] (cf. Theorem 0.9).

By (34), the (non-shifted) Poincaré series $E({\mathfrak{M}}_{d,\chi},q)$ is a
sum of terms of the form

(35)
$q^{d^{2}/2}\prod_{i=1}^{\ell}\frac{q^{k_{i}/2}}{1-q^{k_{i}}}q^{-k_{i}(d_{i}^{2}+1)/2}\mathit{IE}(M_{d_{i},\chi_{i}},q^{k_{i}})$

with

$\ell,\,k_{i}\geq
1\,,\quad\sum_{i=1}^{\ell}k_{i}\chi_{i}=\chi\,,\quad\sum_{i=1}^{\ell}k_{i}d_{i}=d\,,\quad(d_{i},\chi_{i})\in\Lambda_{\mu}\,.$

For any such term

$m=\gcd(d,\chi)=\sum_{i=1}^{\ell}k_{i}\gcd(d_{i},\chi_{i})\geq\sum_{i=1}^{\ell}k_{i}\,.$

Hence the denominator $\prod_{i=1}^{\ell}(1-q^{k_{i}})$ divides
$\prod_{j=1}^{m}(1-q^{j})$ (for example by using the fact that $q$-multinomial
coefficients are polynomials in $q$), so we can write
$E({\mathfrak{M}}_{d,\chi},q)$ in the form stated by the theorem with
$A_{d,m}(q)$ a Laurent polynomial in $q$. It remains to show that $A_{d,m}$ is
actually a polynomial and that it has the specified degree, specified constant
term and specified leading coefficient.

We first observe that terms (35) only contribute to the $q^{s}$ coefficient of
$A_{d,m}(q)$ for

(36)
$s\geq\frac{d^{2}}{2}+\sum_{i=1}^{\ell}\left(\frac{k_{i}}{2}-\frac{k_{i}(d_{i}^{2}+1)}{2}\right)=\frac{d^{2}}{2}-\sum_{i=1}^{\ell}\frac{k_{i}d_{i}^{2}}{2}\geq\frac{d^{2}}{2}-d\sum_{i=1}^{\ell}\frac{k_{i}d_{i}}{2}=0\,.$

This shows that $A_{d,\chi}(q)$ is a polynomial. Equality in (36) holds only
if $\ell=1$, $k_{1}=1$ and $(d_{1},\chi_{1})=(d,\chi)$, so only this term
(which we will refer to as the “main term”) contributes to the constant
coefficient. Similarly we can bound the degree of the rational function and
find that only the main term contributes to the leading coefficient. The
contribution of the main term to $E({\mathfrak{M}}_{d,\chi},q)$ is

$\frac{1}{1-q}\mathit{IE}(M_{d,\chi},q)=\frac{\mathit{IE}(M_{d,\chi},q)(1-q^{2})(1-q^{3})\ldots(1-q^{m})}{(1-q)(1-q^{2})\ldots(1-q^{m})}$

which makes it clear that the constant coefficient is 1 and the top
coefficient is $(-1)^{m-1}$. ∎

######  Remark 4.12.

Note that degree of the rational function on the right hand side is equal to
the dimension $\dim{\mathfrak{M}}_{d,\chi}=d^{2}$. This type of equality need
not be true for arbitrary smooth quotient stacks ${\mathfrak{X}}=[X/G]$.

The BPS integrality formula provides control of the cohomology in small
degrees, and can be used to prove a free generation result for the moduli
stacks analogous to Theorem 1.9 (ii).

######  Theorem 4.13.

The homomorphism ${\mathbb{D}}_{\alpha}\rightarrow
H^{*}({\mathfrak{M}}_{d,\chi})$ defines an isomorphism up to degree $2d-4$.

###### Proof.

We already showed that the homomorphism ${\mathbb{D}}_{\alpha}\to
H^{*}({\mathfrak{M}}_{d,\chi})$ is surjective in Theorem 4.4, so it is enough
to prove that the dimensions agree up to degree $2d-4$. The dimension $\dim
H^{2i}({\mathfrak{M}}_{d,\chi})$ is the $q^{i}$ coefficient of
$E({\mathfrak{M}}_{d,\chi},q)$. We follow the proof of Theorem 4.11 and argue
that only the main term contributes to the coefficient of $q^{i}$ with $i\leq
d-2$ by strengthening the inequality (36). If either $\ell\geq 2$ or
$k_{1}\geq 2$ it is easy to see that

$s\geq\frac{d^{2}}{2}-\frac{(d-1)^{2}}{2}-\frac{1^{2}}{2}=d-1\,,$

showing that indeed the other terms only contribute in higher degrees. Hence,
for $i\leq d-2$ we have the following chain of equalities that we will justify
below:

$\displaystyle\dim H^{2i}({\mathfrak{M}}_{d,\chi})$
$\displaystyle=[q^{i}]E({\mathfrak{M}}_{d,\chi},q)=[q^{i}]\frac{\mathit{IE}(M_{d,\chi},q)}{1-q}=\dim_{2i}H^{*}(M_{d,1})[u]$
$\displaystyle=\dim_{2i}{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}[u]=\dim_{2i}{\mathbb{D}}_{\alpha}\,.$

We have used $[q^{i}]$ to denote the $q^{i}$-coefficient in a series and
$\dim_{2i}$ to denote the dimension of the $2i$-graded part of some graded
algebra. The formal variable $u$ has degree 2. The first equality is by
definition, the second comes from the previous observation that only the main
term contributes for such degrees, the third uses the $\chi$-independence
theorem of Maulik–Shen ([MS23, Theorem 0.1] or Theorem 0.9), the fourth uses
the freeness for the good moduli spaces ([PS23, Theorem 0.2] or Theorem 1.9),
and finally the last equality uses the isomorphism
${\mathbb{D}}_{\alpha}\simeq{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}[u]$ from
Proposition 1.13. ∎

We also have an analogue of Theorem 1.9 (i) for stacks by following
essentially the same strategy as in [PS23].

######  Theorem 4.14.

The algebra $H^{*}({\mathfrak{M}}_{d,\chi})$ is generated by the $3d$ classes

$c_{k+1}(0),\,c_{k}(1),\,c_{k-1}(2)\in
H^{2k}({\mathfrak{M}}_{d,\chi}),\;\;1\leq k\leq d.$

###### Proof.

See Step 1 in [PS23, Section 2.3]. Note that when $\chi=0$ we are not allowed
to use the relations $R_{1}$ (in the notation of loc. cit.) as the inequality
(13) is not satisfied, but this can easily be avoided. ∎

Note that this result is weaker than Theorem 1.9 (i), in the sense that we
require more generators and that we do not claim that this is a minimal
generating set. This is due to the fact that the leading terms analyzed in
Steps 2–4 of [PS23, Section 2.3] are not necessarily non-zero if $\chi=d$ or
$\chi=d/2$ (see for instance the formula for $\det(\mathsf{M})$ in loc. cit.).

##  5\. Computation of the cohomology rings

In this section we compute all the cohomology rings in Theorem 0.2. By the
symmetries before Theorem 1.7, it suffices to restrict our attention to
$0\leq\chi\leq{d}/{2}$. We shall use the normalized classes $c_{k}(j)$ in
Definition 1.6 to present the rings, as explained in Remark 1.10.

###  5.1. Strategy of the computation.

We start by outlining a detailed strategy for the computation, combining the
results and tools we developed in the previous sections. All our computations
are implemented in the software Macaulay2 [GS].

  1. (i)

By Theorem 1.3, the normalized tautological classes $c_{k}(j)$ generate the
cohomology rings of ${\mathfrak{M}}_{d,\chi}$ and $M_{d,\chi}$. For the moduli
spaces $M_{d,\chi}$, classes in cohomological degrees $\leq 2d-4$ generate
according to Theorem 1.9 (i); for the stacks $\mathfrak{M}_{d,\chi}$ classes
in degrees $\leq 2d$ generate the cohomology, see Theorem 4.14. The other
normalized classes can be expressed in terms of these generators via the
relations in ${\mathfrak{I}}_{d,\chi}^{{\mathsf{GMR}},1,\chi^{\prime}}$.

Let $R$ be a free polynomial ring in these generators, with correct
cohomological degrees specified for each. We form an ideal $I$ to which we
shall add geometric relations step by step and take its quotient. For now, we
set $I=(0)$.

  2. (ii)

Compute the Poincaré series of the quotient $R/I$ and compare it with the
Poincaré series of the moduli space $M_{d,\chi}$ (known in small degrees, see
for example [LP93a, CC16, CC15]) or the moduli stack $\mathfrak{M}_{d,\chi}$
(obtained from BPS integrality). Identify the first cohomological degree, say
$\ell$, where the two series do not match. This is where the first missing
relations occur.

  3. (iii)

Apply Virasoro operators to generators of $I$ to obtain relations in degree
$\ell$. Using Step (i), reduce the new relations to expressions in terms of
the ring generators only, and add these to the ideal $I$.

  4. (iv)

If Step (iii) does not give all the missing relations in degree $\ell$,
produce primitive relations in degree $\ell$. Note that we do not need
primitive relations of degrees less than $\ell$ according to Theorem 3.13,
since we have already performed Step (iii). Reduce the new relations and add
them to the ideal $I$.

  5. (v)

Repeat Steps (ii)-(iv) above, until the Poincaré series match in all degrees,
meaning that we have obtained all the tautological relations141414Note that
$R$ is Noetherian and the ideal of tautological relations is finitely
generated., or conclude that the geometric relations are not complete.

  6. (vi)

(Optional) Use the command minimalPresentation to obtain a minimal
presentation of the cohomology ring and the ideal $I$.

######  Remark 5.1.

Although the strategy above makes use of the known Poincaré series [LP93a,
CC16, CC15] of the moduli spaces $M_{d,\chi}$, we remark that our results have
an independent nature and do not depend on these numerical data in any
essential way. Indeed, since $M_{d,\chi}$ is smooth and projective when
$\gcd(d,\chi)=1$, once we have an ideal $I$ of geometric relations such that
the quotient ring $R/I$ is Gorenstein with the correct dimension, Poincaré
duality would force an isomorphism $H^{*}(M_{d,\chi})\simeq R/I$. The fact
that we do have enough relations to obtain the cohomology rings is precisely
the content of Theorem 0.2 (for the moduli spaces). One then follows Steps
(ii)-(v) to treat the moduli stacks. We chose to present our strategy as above
since it is in practice much more algorithmic.

###  5.2. The cohomology rings.

In this section, we describe the computation of
$H^{*}(\mathfrak{M}_{2,0},\mathbb{Q})$ in detail following the above strategy.
For the other moduli stacks (or spaces), we merely list key information such
as Poincaré series, numbers of generators and relations, etc. for their
cohomology rings, as the computations are much more involved. All
presentations of the cohomology rings, including the partial one for
$\mathfrak{M}_{5,0}$, are uploaded to the fourth author’s website

https://github.com/Weite-Pi/weitepi.github.io

in a folder named cohomology rings.

######  Remark 5.2.

The moduli space $M_{1,0}$ is isomorphic to $\mathbb{P}^{2}$, whose cohomology
is well-known and can be obtained as in Example 2.5. The moduli spaces
$M_{2,1}$ and $M_{3,1}$ are also classical [LP93a]. The cohomology of the
moduli space $M_{4,1}$ is first computed in [CM17] using birational geometry,
and translated to a presentation in terms of normalized classes $c_{k}(j)$ in
[KPS23]. As explained in the previous remark, the cohomology rings of these
moduli spaces can all be obtained directly using geometric relations, and we
omit the details for them in the following.

####  5.2.1. The moduli stack $\mathfrak{M}_{2,0}$.

The cohomology of the stack $\mathfrak{M}_{2,0}$ is generated by the
normalized classes in degrees $\leq 4$. We thus input the command

R = QQ[c02,c11,c20,c12,c21,c30, Degrees => {2,2,2,4,4,4}]

in Macaulay2 to get a free polynomial ring. Its Poincaré series has first
terms

$E(R,q)=1+3q+9q^{2}+19q^{3}+39q^{4}+69q^{5}+119q^{6}+\cdots.$

On the other hand, the correct Poincaré series for $\mathfrak{M}_{2,0}$ can be
obtained by BPS integrality:

$\displaystyle E(\mathfrak{M}_{2,0},q)$
$\displaystyle=\frac{1+q+2q^{2}+2q^{3}+3q^{4}+q^{5}-q^{7}}{\left(1-q\right)\left(1-q^{2}\right)}$
$\displaystyle=1+2q+5q^{2}+8q^{3}+14q^{4}+18q^{5}+24q^{6}+\cdots.$

Thus the first relation occurs in cohomological degree 2 (note that this is
twice the exponent of $q$ since the latter records algebraic degrees). We
produce a Mumford relation in this degree:

$\mathbf{r}_{1}=c_{2}(0)-\frac{1}{8}c_{0}(2).$

One can also use ${\mathsf{GMR}}$ with $M_{1,\chi^{\prime}}$, which produces
the same relation. Setting $I$ to be the ideal generated by this relation, we
compute the Poincaré series

$E(R/I,q)=1+2q+6q^{2}+10q^{3}+20q^{4}+30q^{5}+50q^{6}+\cdots.$

Comparing with $E(\mathfrak{M}_{2,0},q)$, we see that the next relation occurs
in (cohomological) degree 4. We apply the Virasoro operator $\mathsf{L}_{1}$
to the relation $c_{2}(0)-\frac{1}{8}c_{0}(2)$, which gives a new relation

$\mathbf{r}_{2}=-\frac{1}{4}c_{1}(2)+2c_{3}(0).$

Adding this to the ideal $I$, we have

$E(R/I,q)=1+2q+5q^{2}+8q^{3}+14q^{4}+20q^{5}+30q^{6}+\cdots.$

The next relations thus occur in degree 10, and there are two of them. We
apply the Virasoro operators to the two relations in $I$; this time, however,
they do not give anything new, i.e. the relations we obtain still lie in $I$.
We proceed by producing ${\mathsf{GMR}}$ with $M_{1,\chi^{\prime}}$, and this
does give two new relations. After simplifying, they are

$\displaystyle\mathbf{r}_{3}=c_{1}(1)c_{2}(0)^{4}-2c_{2}(0)^{3}c_{3}(0),$
$\displaystyle\mathbf{r}_{4}=c_{1}(1)^{2}c_{2}(0)^{3}+16c_{2}(0)^{5}+2c_{2}(0)^{3}c_{2}(1)-6c_{1}(1)c_{2}(0)^{2}c_{3}(0)+6c_{2}(0)c_{3}(0)^{2}.$

Adding these to $I$, we have

$E(R/I,q)=1+2q+5q^{2}+8q^{3}+14q^{4}+18q^{5}+26q^{6}+\cdots.$

The next two missing relations lie in degree 12. We apply Virasoro operators
to the four generators in $I$, and this gives two new relations
$\mathbf{r}_{5}=\mathsf{L}_{1}(\mathbf{r}_{3})$ and
$\mathbf{r}_{6}=\mathsf{L}_{1}(\mathbf{r}_{4})$. Adding these to the ideal and
computing the Poincaré series, we found that151515Macaulay2 is able to reduce
Poincaré series to this rational form using the command reduceHilbert.

$E(R/I,q)=\frac{1+q+2q^{2}+2q^{3}+3q^{4}+q^{5}-q^{7}}{\left(1-q\right)\left(1-q^{2}\right)},$

matching the correct Poincaré series of $\mathfrak{M}_{2,0}$. Thus we have
found all the relations and

$H^{*}(\mathfrak{M}_{2,0},\mathbb{Q})\simeq R/I$

where
$I=\langle\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3},\mathbf{r}_{4},\mathbf{r}_{5},\mathbf{r}_{6}\rangle$.

We can further trim the quotient ring and obtain a minimal presentation of
$H^{*}(\mathfrak{M}_{2,0},\mathbb{Q})$. In the trimmed form, it has four
tautological generators

$c_{1}(1),\>c_{2}(0),\>c_{2}(1),\>c_{3}(0),$

and four relations, two in degree 10 and two in degree 12.

####  5.2.2. The moduli stack $\mathfrak{M}_{3,0}$.

The Poincaré series of $\mathfrak{M}_{3,0}$ is as follows:

$\displaystyle E(\mathfrak{M}_{3,0},q)$
$\displaystyle=\frac{1}{\left(1-q\right)\left(1-q^{2}\right)\left(1-q^{3}\right)}\cdot(1+2q+3q^{2}+3q^{3}+3q^{4}+4q^{5}+7q^{6}$
$\displaystyle\qquad\qquad+6q^{7}+5q^{8}+q^{9}-4q^{10}-6q^{11}-2q^{12}+q^{13}+2q^{14}+q^{15})$
$\displaystyle=1+3q+7q^{2}+13q^{3}+22q^{4}+35q^{5}+55q^{6}+79q^{7}+111q^{8}+\cdots.$

We obtain $H^{*}(\mathfrak{M}_{3,0},\mathbb{Q})$ following the same algorithm
as for $\mathfrak{M}_{2,0}$. In the trimmed form, it has 7 generators

$c_{0}(2),\>c_{1}(1),\>c_{2}(0),\>c_{1}(2),\>c_{2}(1),\>c_{2}(2),\>c_{3}(1).$

There are in total 16 relations among these generators:

degrees | $H^{4}$ | $H^{6}$ | $H^{8}$ | $H^{14}$ | $H^{16}$ | $H^{18}$ | $H^{20}$
---|---|---|---|---|---|---|---
# of rel. | 1 | 2 | 2 | 2 | 3 | 4 | 2

####  5.2.3. The moduli stack $\mathfrak{M}_{4,2}$.

The Poincaré series of $\mathfrak{M}_{4,2}$ is as follows:

$\displaystyle E(\mathfrak{M}_{4,2},q)$
$\displaystyle=\frac{1}{(1-q)(1-q^{2})}\cdot(1+2q+5q^{2}+8q^{3}+9q^{4}+6q^{5}+5q^{6}+4q^{7}+5q^{8}$
$\displaystyle\qquad\;\;+5q^{9}+6q^{10}+4q^{11}+3q^{12}-3q^{14}-8q^{15}-8q^{16}-5q^{17}-2q^{18}-q^{19})$
$\displaystyle=1+3q+9q^{2}+19q^{3}+34q^{4}+50q^{5}+70q^{6}+90q^{7}+115q^{8}+\cdots.$

In the trimmed form, the cohomology ring
$H^{*}(\mathfrak{M}_{4,2},\mathbb{Q})$ has 7 generators

$c_{0}(2),\>c_{1}(1),\>c_{2}(0),\>c_{1}(2),\>c_{2}(1),\>c_{3}(0),\>c_{4}(0).$

There are in total 19 relations among these generators:

degrees | $H^{6}$ | $H^{8}$ | $H^{10}$ | $H^{12}$ | $H^{22}$ | $H^{24}$ | $H^{26}$ | $H^{28}$ | $H^{30}$
---|---|---|---|---|---|---|---|---|---
# of rel. | 1 | 5 | 5 | 1 | 2 | 2 | 1 | 1 | 1

####  5.2.4. The moduli stack $\mathfrak{M}_{4,0}$.

The Poincaré series of $\mathfrak{M}_{4,0}$ is as follows:

$\displaystyle E(\mathfrak{M}_{4,0},q)$
$\displaystyle=\frac{1}{(1-q)^{2}(1-q^{2})(1-q^{4})}\cdot(1+q+3q^{2}+4q^{3}+3q^{4}+q^{5}+3q^{6}+2q^{7}$
$\displaystyle\quad+5q^{8}+6q^{9}+7q^{10}+4q^{11}+3q^{12}-6q^{13}-11q^{14}-13q^{15}-6q^{16}+3q^{17}$
$\displaystyle\quad+10q^{18}+10q^{19}+4q^{20}-2q^{21}-3q^{22}-q^{23}-q^{24})$
$\displaystyle=1+3q+9q^{2}+20q^{3}+39q^{4}+65q^{5}+105q^{6}+158q^{7}+234q^{8}+\cdots.$

In the trimmed form, the cohomology ring
$H^{*}(\mathfrak{M}_{4,0},\mathbb{Q})$ has 10 generators

$c_{0}(2),\>c_{1}(1),\>c_{2}(0),\>c_{1}(2),\>c_{2}(1),\>c_{3}(0),\>c_{3}(1),\>c_{4}(0),\>c_{4}(1),\>c_{5}(0).$

There are in total 40 relations among these generators:

degrees | $H^{6}$ | $H^{8}$ | $H^{10}$ | $H^{12}$ | $H^{18}$ | $H^{20}$ | $H^{22}$ | $H^{24}$ | $H^{26}$ | $H^{28}$
---|---|---|---|---|---|---|---|---|---|---
# of rel. | 1 | 5 | 5 | 2 | 2 | 2 | 4 | 6 | 8 | 5

####  5.2.5. The moduli space $M_{5,1}$.

The Poincaré series of $M_{5,1}$ have been computed by various methods, see
for example [Yua14, CC16, Mai13]:

$\displaystyle E(M_{5,1},q)$
$\displaystyle=1+2q+6q^{2}+13q^{3}+26q^{4}+45q^{5}+68q^{6}+87q^{7}+100q^{8}+107q^{9}+111q^{10}$
$\displaystyle\quad+112q^{11}+113q^{12}+113q^{13}+113q^{14}+112q^{15}+111q^{16}+107q^{17}+100q^{18}$
$\displaystyle\quad+87q^{19}+68q^{20}+45q^{21}+26q^{22}+13q^{23}+6q^{24}+2q^{25}+q^{26}.$

As we mentioned in Section 0.2, the geometric relations
${I}^{\mathsf{Geom}}_{5,1}$ turn out to be not complete. Indeed, our algorithm
stops when we try to find a new relation in degree 36. On the other hand,
imposing all geometric relations does reduce the top dimension to one:

$\displaystyle E(R/{I}^{\mathsf{Geom}}_{5,1},q)$
$\displaystyle=1+2q+6q^{2}+13q^{3}+26q^{4}+45q^{5}+68q^{6}+87q^{7}+100q^{8}+107q^{9}+111q^{10}$
$\displaystyle\quad+112q^{11}+113q^{12}+113q^{13}+113q^{14}+112q^{15}+111q^{16}+107q^{17}+101q^{18}$
$\displaystyle\quad+91q^{19}+78q^{20}+\cdots+q^{26}.$

Recall that $M_{5,1}$ is a smooth projective variety [LP93a]. In particular,
the cohomology ring $H^{*}(M_{5,1},\mathbb{Q})$ satisfies Poincaré duality.
Thus once the top dimension is reduced to one, we can use the Poincaré pairing
to find the remaining relations. This completely determines the ring
structure.

The generators for $M_{5,1}$ are given by Theorem 1.3. There are in total 34
relations as follows. The cross signs indicate three relations that are not
geometric.

degrees | $H^{10}$ | $H^{12}$ | $H^{14}$ | $H^{30}$ | $H^{34}$ | $H^{36}$ | $H^{38}$ | $H^{40}$
---|---|---|---|---|---|---|---|---
# of rel. | 3 | 12 | 13 | $1$ | $1$ | $1^{\times}$ | $2^{\times}$ | $1$

####  5.2.6. The moduli space $M_{5,2}$.

The Poincaré series and ring generators of $M_{5,2}$ are the same as those of
$M_{5,1}$. This time the geometric relations are complete, namely
$I^{\mathsf{Geom}}_{5,2}=I_{5,2}$ and also
${\mathfrak{I}}^{\mathsf{Geom}}_{5,2}={\mathfrak{I}}_{5,2}$ on the stack
level. There are in total 41 relations for the moduli space:

degrees | $H^{10}$ | $H^{12}$ | $H^{14}$ | $H^{16}$ | $H^{18}$ | $H^{28}$ | $H^{30}$ | $H^{32}$ | $H^{34}$ | $H^{36}$ | $H^{38}$ | $H^{40}$
---|---|---|---|---|---|---|---|---|---|---|---|---
# of rel. | 3 | 12 | 13 | 2 | 1 | 1 | $1$ | $1$ | 2 | $3$ | $1$ | $1$

######  Remark 5.3.

Our results on the cohomology of $M_{5,1}$ and $M_{5,2}$ give a different and
explicit proof for the first non-trivial case of Theorem 1.7. Also as we
discussed in Remark 2.12, one can ask whether $I^{{\mathsf{GMR}},5,2}_{5,1}$
will produce the missing tautological relations for $M_{5,1}$.

####  5.2.7. The moduli stack $\mathfrak{M}_{5,0}$.

Finally, we determine the cohomology ring of the stack $\mathfrak{M}_{5,0}$ up
to cohomological degree 32. We did not complete this for a purely
computational reason, as the current ring structure is already quite
complicated, and it takes an exceedingly long time to compute Poincaré series
or to reduce the relations. We hope this partial result would still be
interesting and useful for future research.

First, we list the first terms in the Poincaré series of $\mathfrak{M}_{5,0}$:

$\displaystyle E(\mathfrak{M}_{5,0},q)$
$\displaystyle=1+3q+9q^{2}+22q^{3}+49q^{4}+98q^{5}+180q^{6}+302q^{7}+476q^{8}$
$\displaystyle+713q^{9}+1032q^{10}+1449q^{11}+1994q^{12}+2690q^{13}+3575q^{14}$
$\displaystyle+4679q^{15}+6047q^{16}+7706q^{17}+9696q^{18}+12027q^{19}+14731q^{20}+\cdots$

Our computation shows that the geometric relations are complete up to
cohomological degree 32. The cohomology ring $H^{*}(\mathfrak{M}_{5,0})$ has
13 generators

$c_{0}(2),\>c_{1}(1),\>c_{2}(0),\>c_{1}(2),\>c_{2}(1),\>c_{3}(0),\>c_{2}(2),\>c_{3}(1),\>c_{4}(0),\>c_{3}(2),\>c_{4}(1),\>c_{4}(2),\>c_{5}(1).$

There are in total 56 relations up to degree 32:

degrees | $H^{8}$ | $H^{10}$ | $H^{12}$ | $H^{14}$ | $H^{16}$ | $H^{18}$ | $H^{22}$ | $H^{24}$ | $H^{26}$ | $H^{28}$ | $H^{30}$ | $H^{32}$ | $\cdots$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
# of rel. | 1 | 3 | 8 | 11 | 5 | 1 | 2 | 2 | 2 | 4 | 7 | 10 | $\cdots$

The next relations occur in degree 34, and there are 14 of them. To find these
relations, one can start by applying the Virasoro operators to the existing
ones.

##  6\. $P=C$ and other conjectures

In this final section, we formulate the strengthened version of the $P=C$
conjecture and verify several predictions on the cohomology rings of the
moduli spaces (mostly $M_{5,\chi}$) in Theorem 0.2. Again, it suffices to
verify each of the conjectures for $0\leq\chi\leq{d}/{2}$.

###  6.1. The $P=C$ conjecture.

####  6.1.1. The perverse filtration

We start by recalling the construction of the perverse filtration [dC16,
Lecture 5]; see also [KPS23, Section 1.1]. Let $f:X\to Y$ be a proper morphism
between irreducible varieties. For simplicity, we assume that $X$ and $Y$ are
both smooth and projective. The perverse truncation functors [BBD82] filter
the derived push-forward complex $Rf_{*}\mathbb{Q}_{X}$, thus inducing an
increasing filtration

$P_{0}H^{*}(X)\subset P_{1}H^{*}(X)\subset\cdots\subset H^{*}(X,\mathbb{Q})$

called the perverse filtration. Intuitively, it measures the complexity of the
fibers of the proper map $f:X\to Y$.

We recall the following useful properties of the perverse filtration:

  1. (i)

Let $\eta$ be an ample class on $Y$, and denote by $\xi:=f^{*}\eta$ its pull-
back to $X$. Then the perverse filtration on $H^{*}(X,\mathbb{Q})$ is
completely determined by the multiplication by $\xi$ operator; see [dCM05,
Proposition 5.2.4] and [KPS23, Proposition 1.1] for the explicit formula.

  2. (ii)

Let $r$ be the defect of semismallness of $f$:

$r:=\dim X\times_{Y}X-\dim X$

When $f$ is equidimensional, this is simply the relative dimension $\dim
X-\dim Y$. Then the perverse filtration $P_{\bullet}H^{*}(X,\mathbb{Q})$
stabilizes in $2r$ steps, i.e. $P_{2r}H^{*}(X)=H^{*}(X)$.

  3. (iii)

The graded pieces $\mathrm{gr}^{P}_{\bullet}H^{*}(X)$ of the perverse
filtration satisfy two Hard Lefschetz type symmetries [dCM05, Theorem 2.1.4].
They arise respectively from the pull-back of an ample class on $Y$ and a
relative ample class on $X$.

Consider the moduli space $M_{d,\chi}$ with $\gcd(d,\chi)=1$ and the
Hilbert–Chow map

$h:M_{d,\chi}\longrightarrow|d\cdot H|.$

This is a proper map with generic fibers being abelian varieties. Its topology
is encoded in the perverse filtration
$P_{\bullet}H^{*}(M_{d,\chi},\mathbb{Q})$. This is one side of the $P=C$
conjecture.

####  6.1.2. The Chern filtration

We turn to the Chern side next. Recall that the (local) Chern filtration
$C_{\bullet}H^{\leq 2d-4}(M_{d,\chi})$ was defined in [KPS23, Section 0.2] for
coprime $M_{d,\chi}$. By definition, it involves only the normalized
tautological generators in Theorem 1.9 (i). In order to extend this to a
filtration on the entire cohomology rings, it is natural to include all
normalized tautological classes. We thus define the (global) Chern filtration

(37)
$C_{k}H^{*}(M_{d,\chi})\coloneqq\mathrm{span}\left\\{\prod_{i=1}^{s}c_{k_{i}}(j_{i})\,\Bigg{|}\,\sum_{i=1}^{s}k_{i}\leq
k\right\\}\subset H^{*}(M_{d,\chi},\mathbb{Q})$

where we impose no restriction on the cohomological degree, and
$c_{k_{i}}(j_{i})$ is allowed to be any normalized class.161616This corrects
the mismatch between perversity and the Chern degrees in [KPS23, Remark 2.10].
Equivalently, this is the smallest multiplicative filtration where each class
$c_{k}(j)$ lies in the $k$-th step of the filtration.

The definition above has the advantage of being explicit and computable, but
apparently relies on the normalization in Section 1.2. Recall that the
normalized classes are closely related to the weight zero descendent algebra
via the identification
${\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]\simeq{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$,
where $\alpha=(d,\chi)$ as usual. We now explain how the latter can be used to
give a more canonical definition of the Chern filtration, without any choice
of normalization. To motivate this, we note that the previous definition (37)
can be viewed as being induced from a parallel Chern filtration
$C_{\bullet}{\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]$ on the free
polynomial ring, via the surjective map

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

We can also define the Chern filtration
$C_{\bullet}H^{*}({\mathfrak{M}}_{d,\chi})$ for the stacks in the same way,
using a surjective map from
$C_{\bullet}{\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]$ for any
$(d,\chi)$.

######  Definition 6.1.

The Chern filtration
$\widetilde{C}_{\bullet}{\mathbb{D}}_{\alpha}\subset\mathbb{D}_{\alpha}$ is
the minimal multiplicative filtration such that
${\mathrm{ch}}_{k}(H^{j})\in\widetilde{C}_{k-j+1}\,{\mathbb{D}}_{\alpha}$ for
any $k>0$ and $j=0,1,2$. The Chern filtration on the weight zero descendent
algebra ${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$ is defined by restriction,
i.e.

$\widetilde{C}_{\bullet}{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}=\widetilde{C}_{\bullet}{\mathbb{D}}_{\alpha}\cap{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\,.$

######  Remark 6.2.

Since
$\xi({\mathrm{ch}}_{k}(H^{j}))=p_{*}\big{(}{\mathrm{ch}}_{k-j+2}({\mathcal{F}})\cdot
q^{*}H^{j}\big{)}$, the Chern filtration index $(k-j+1)$ is one less than the
Chern character degree of the universal sheaf we take in the realization. The
shift by one can be motivated by the codimension of the sheaves we
parametrize.

######  Remark 6.3.

The Chern filtrations are preserved under the isomorphism

$\widetilde{C}_{\bullet}{\mathbb{D}}_{e^{\rho}\cdot\alpha}\xrightarrow{{\mathsf{F}}_{\rho}}\widetilde{C}_{\bullet}{\mathbb{D}}_{\alpha}$

from Section 1.2.2. Indeed, we have
${\mathsf{F}}_{\rho}({\mathrm{ch}}_{k}(H^{j}))={\mathrm{ch}}_{k}(H^{j}\cdot
e^{\rho})\in\widetilde{C}_{k-j+1}{\mathbb{D}}_{\alpha}$, and the
multiplicativity shows the inclusion
${\mathsf{F}}_{\rho}(\widetilde{C}_{\bullet})\subseteq\widetilde{C}_{\bullet}$.
The other inclusion follows from the same argument applied to
${\mathsf{F}}_{-\rho}$. The same statement holds for the weight zero
descendent algebra.

######  Definition 6.4.

We define the Chern filtration on the cohomology of the moduli stacks and
spaces to be

$\widetilde{C}_{\bullet}H^{*}({\mathfrak{M}}_{d,\chi})\coloneqq\textup{Im}(\widetilde{C}_{\bullet}{\mathbb{D}}_{\alpha}\overset{\xi}{\longrightarrow}H^{*}({\mathfrak{M}}_{d,\chi})),\quad\widetilde{C}_{\bullet}H^{*}(M_{d,\chi})\coloneqq\textup{Im}(\widetilde{C}_{\bullet}{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}\overset{\xi}{\longrightarrow}H^{*}(M_{d,\chi}))$

where $\gcd(d,\chi)=1$ is assumed for the moduli space.

The next proposition justifies the same choice of terminology for the above
definitions.

######  Proposition 6.5.

The two definitions of the Chern filtration match, i.e.

$C_{\bullet}H^{*}({\mathfrak{M}}_{d,\chi})=\widetilde{C}_{\bullet}H^{*}({\mathfrak{M}}_{d,\chi})\quad\textup{and}\quad
C_{\bullet}H^{*}(M_{d,\chi})=\widetilde{C}_{\bullet}H^{*}(M_{d,\chi})$

where $\gcd(d,\chi)=1$ is assumed for the moduli space.

###### Proof.

It suffices to show that the two filtrations match under the identifications
of the formal algebras as in Section 1.2.3. Consider the commutative diagram
of ring homomorphisms

${{\mathbb{Q}}[c_{0}(2),c_{1}(1),c_{2}(0),c_{1}(2),\cdots]}$${{\mathbb{D}}_{\alpha}}$${{\mathbb{Q}}[c_{0}(2),c_{2}(0),c_{1}(2),\cdots]}$${{\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}.}$$\scriptstyle{\sim}$$\scriptstyle{{\eta}^{\prime}}$$\scriptstyle{\eta}$$\scriptstyle{\sim}$

The left and the right columns are equipped with the $C_{\bullet}$ and
$\widetilde{C}_{\bullet}$ filtrations, respectively. The top row isomorphism
identifies the two Chern filtrations $C_{\bullet}$ and
$\widetilde{C}_{\bullet}$ by Remark 6.3. This proves the statement for
${\mathfrak{M}}_{d,\chi}$.

Now we consider the moduli space $M_{d,\chi}$ which corresponds to the bottom
row. On the left hand side, it is clear that
${\eta}^{\prime}(C_{\bullet})=C_{\bullet}$ since ${\eta}^{\prime}$ simply sets
$c_{1}(1)=0$. Therefore, we are left with showing that
$\eta(\widetilde{C}_{\bullet})=\widetilde{C}_{\bullet}$ on the right hand
side. Since $\eta$ restricts to the identity on
${\mathbb{D}}_{\alpha,{\mathrm{wt}_{0}}}$, we have
$\eta(\widetilde{C}_{\bullet})\supseteq\widetilde{C}_{\bullet}$. On the other
hand, recall the formula

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

It is clear from the definition of ${\mathsf{R}}_{-1}$ and the
$\widetilde{C}_{\bullet}$ filtration that
${\mathsf{R}}_{-1}(\widetilde{C}_{\bullet})\subseteq\widetilde{C}_{\bullet-1}$
and
${\mathrm{ch}}_{1}(H)\widetilde{C}_{\bullet}\subseteq\widetilde{C}_{\bullet+1}$,
since ${\mathrm{ch}}_{1}(H)\in\widetilde{C}_{1}$. We conclude that
$\eta(\widetilde{C}_{\bullet})\subseteq\widetilde{C}_{\bullet}$, which
completes the proof. ∎

We expect the Chern filtration $C_{\bullet}H^{*}({\mathfrak{M}}_{d,\chi})$ to
play a role in a stacky version of the $P=C$ conjecture; see Theorem 6.9 for
the numerical evidence.

####  6.1.3. The conjecture and the result

We are now ready to state the (strengthened) $P=C$ conjecture.

######  Conjecture 6.6.

For coprime $d\geq 1$ and $\chi\in\mathbb{Z}$, we have

$P_{\bullet}H^{*}(M_{d,\chi},\mathbb{Q})=C_{\bullet}H^{*}(M_{d,\chi},\mathbb{Q}).$

It was shown in [MSY23, Theorem 0.6] that the Chern filtration is contained in
the perverse filtration in small cohomological degrees, i.e.

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

In addition, the equality between the two filtrations up to cohomological
degree 4 is shown in [Yua23a].

######  Remark 6.7.

Before presenting more evidence, we discuss a number of structural
implications of Conjecture 6.6 on both sides of the equality.

  1. (i)

The perverse filtration is in general not multiplicative, meaning that we do
not necessarily have

$P_{k}H^{*}(X,\mathbb{Q})\times
P_{k^{\prime}}H^{*}(X,\mathbb{Q})\overset{\cup}{\longrightarrow}P_{k+k^{\prime}}H^{*}(X,\mathbb{Q}).$

See [dC16, Exercise 5.6.8] for a counterexample, and [dCMS22, Theorem 0.6] on
how the $P=W$ conjecture can be reduced to a multiplicativity statement. On
the other hand, the (global) Chern filtration is multiplicative by definition.
Thus Conjecture 6.6 would imply that the perverse filtration on $M_{d,\chi}$
is multiplicative as well.

  2. (ii)

The Hilbert–Chow morphism is equidimensional, and thus
$P_{\bullet}H^{*}(M_{d,\chi})$ stabilizes in $2g$ steps, where
$g=\frac{1}{2}(d-1)(d-2)$ is the relative dimension. This implies that the
Chern filtration should also stabilize, i.e.
$C_{2g}H^{*}(M_{d,\chi})=H^{*}(M_{d,\chi})$. Explicitly, every monomial in the
normalized classes $c_{k}(j)$, possibly with Chern degrees $>2g$, should be
expressed by those with Chern degrees $\leq 2g$ in the cohomology ring
$H^{*}(M_{d,\chi},\mathbb{Q})$.

  3. (iii)

According to [MS23, Theorem 0.4], the perverse numbers (i.e. dimensions of
graded pieces of the perverse filtration) of $M_{d,\chi}$ are independent on
the Euler characteristic $\chi$, and satisfy two Hard Lefschetz type
symmetries [dCM05, Theorem 2.1.4]. Thus the graded dimensions of the Chern
filtration on $M_{d,\chi}$ should also be independent on $\chi$ and satisfy
these symmetries. This prediction is particularly non-trivial given Theorem
1.7, which states that the cohomology ring structure of $M_{d,\chi}$ is as
$\chi$-dependent as possible.

  4. (iv)

Also from the equidimensionality of the Hilbert–Chow map and the two Hard
Lefschetz symmetries, the perverse filtration $P_{\bullet}H^{*}(M_{d,\chi})$
is not ‘full’ when the cohomological degree is large. For example, it only has
the highest perversity piece

$0=P_{2g-1}H^{\mathrm{top}}(M_{d,\chi})\subset
P_{2g}H^{\mathrm{top}}(M_{d,\chi})=H^{\mathrm{top}}(M_{d,\chi},\mathbb{Q})$

in top cohomological degree. More generally, we have

$P_{2\ell-1}H^{2(b+\ell)}(M_{d,\chi},\mathbb{Q})=0\quad\textrm{for \,}\ell\geq
1.$

Conjecture 6.6 thus predicts that monomials in $c_{k}(j)$ that lie in
$C_{2\ell-1}H^{2(b+\ell)}(M_{d,\chi})$ should also vanish, providing a wealth
of non-trivial relations in the cohomology ring
$H^{*}(M_{d,\chi},\mathbb{Q})$. For example, the base relations (cf.
Proposition 2.14) are a special case of this prediction since

$\prod_{i=1}^{b+1}c_{k_{i}}(2)\in C_{k}H^{2b+2k+2}(M_{d,\chi})\subseteq
C_{2(k+1)-1}H^{2(b+k+1)}(M_{d,\chi})$

where $k=\sum_{i=1}^{b+1}k_{i}$. It remains to be studied how these
(conjectural) relations determine the ring structure.

Here is our main evidence for the $P=C$ conjecture:

######  Theorem 6.8.

Conjecture 6.6 holds for all moduli spaces $M_{d,\chi}$ with $d\leq 5$ and
$\gcd(d,\chi)=1$. In particular, the perverse filtrations on these moduli
spaces are multiplicative.

###### Proof.

According to [KPS23, Corollary 1.3], the perverse filtration on $M_{d,\chi}$
is characterized by the cup product with the class $\xi=c_{0}(2)$. Thus the
cohomology ring structure completely determines the perverse filtration. The
theorem follows by straightforward verifications of the $P=C$ equality as
explained in [KPS23, Section 2.4]. ∎

We also record the following result towards a stacky $P=C$ conjecture, without
going into details on how the perverse filtration on the stacks [Dav23] is
defined. The proof is again a direct verification in Macaulay2.171717See
Remark 6.11 on how the stacky perverse numbers can be computed via BPS
integrality.

######  Theorem 6.9.

For the moduli stacks
${\mathfrak{M}}_{2,0},\,{\mathfrak{M}}_{3,0},\,{\mathfrak{M}}_{4,2},\,{\mathfrak{M}}_{4,0}$,
the graded dimensions of the perverse filtration and the Chern filtration
coincide in all cohomological degrees.

The same numerical match holds for ${\mathfrak{M}}_{5,0}$ up to cohomological
degree $32$ where we are able to determine the ring structure, cf. Section
5.2.7.

###  6.2. Refined GV/PT/Nekrasov correspondence

As we have discussed in the overview, one major motivation to study
$M_{d,\chi}$, and in particular the perverse filtration
$P_{\bullet}H^{*}(M_{d,\chi})$, arises from its connection to the refined BPS
invariants of local $\mathbb{P}^{2}$, which we write simply
$K_{{\mathbb{P}}^{2}}$. We organize the numerical information of the perverse
filtration in the Laurent polynomial in two variables

$\Omega_{d}(q,t)=(-1)^{d^{2}+1}q^{-g}t^{-b}\sum_{i,j\geq
0}\dim\mathrm{gr}_{i}^{P}\mathit{IH}^{i+j}(M_{d,\chi})q^{i}t^{j}\,,$

where $b=\frac{d(d+3)}{2}$ and $g=\frac{(d-1)(d-2)}{2}$ are the dimension of
the base and fibers of the Hilbert–Chow map, respectively. The right hand side
does not depend on $\chi$ by Theorem 0.9. The two hard Lefschetz symmetries
imply that $\Omega_{d}$ is invariant under the two symmetries
$q\leftrightarrow q^{-1}$ and $t\leftrightarrow t^{-1}$.

We define the refined BPS invariants $N_{d}^{j_{L},j_{R}}$ for $d\geq 1$ and
$j_{L},j_{R}\in\frac{1}{2}{\mathbb{Z}}_{\geq 0}$ to be the unique integers
satisfying the equality

(38)
$\Omega_{d}(q,t)=\sum_{j_{L},j_{R}}(-1)^{2j_{L}+2j_{R}}N_{d}^{j_{L},j_{R}}\chi_{j_{L}}(q)\chi_{j_{R}}(t)$

where

$\chi_{j}(x)=\frac{x^{2j+1}-x^{-2j-1}}{x-x^{-1}}=x^{-2j}+x^{-2j+2}+\ldots+x^{2j-2}+x^{2j}\,.$

The numbers $N_{d}^{j_{L},j_{R}}$ are supposed to be the numbers of BPS
particles with charge $d$ and quantum spin numbers $(j_{L},j_{R})$ in a
certain 5D gauge theory; see for example the discussion in the introductions
of [IKV09] or [CKK14]. These numbers are a refinement of the standard
Gopakumar–Vafa invariants, which conjecturally control the enumerative
geometry (Gromov–Witten or Pandharipande–Thomas invariants) of
$K_{{\mathbb{P}}^{2}}$. Indeed, the definition of Maulik–Toda [MT18] of the
(unrefined) Gopakumar–Vafa invariants $n_{g,d}$ of $K_{{\mathbb{P}}^{2}}$ is

$(-1)^{d^{2}+1}\Omega_{d}(q,-1)=\Omega_{d}(-q,1)=\sum_{g\geq
0}n_{g,d}\big{(}q^{1/2}+q^{-1/2}\big{)}^{2g}\,.$

Choi, Katz and Klemm propose in [CKK14] to use refined stable pairs to define
$N_{d}^{j_{L},j_{R}}$. Comparing their proposal to our definition of
$N_{d}^{j_{L},j_{R}}$ leads to a refined GV/PT correspondence, which we now
formulate precisely.

The (unrefined) Pandharipande–Thomas invariants are integers
$\mathsf{PT}_{d,n}$ which (virtually) count stable pairs on
$K_{{\mathbb{P}}^{2}}$ with discrete invariants $(d,n)$. Pandharipande–Thomas
invariants admits a refinement181818See for example [NO16] or [CKK14]. Our
formal variable $t$ corresponds to $\kappa^{1/2}$ in [NO16] or to
${\mathbb{L}}^{1/2}$ in [CKK14].

$\mathsf{PT}_{d,n}^{\mathsf{ref}}\in{\mathbb{Z}}[t^{\pm 1}]$

symmetric under $t\leftrightarrow t^{-1}$ which specializes to
$\mathsf{PT}_{d,n}$ when $t=1$. The refined PT vertex is a generating series
in formal variables $q,t,Q$:

$Z^{\mathsf{ref}}_{\mathsf{PT}}=\sum_{d\geq
0}Q^{d}\sum_{n\in{\mathbb{Z}}}\mathsf{PT}_{d,n}^{\mathsf{ref}}\cdot(-q)^{n}\,.$

The refined PT vertex $Z^{\mathsf{ref}}_{\mathsf{PT}}$ can be combinatorially
calculated (at least up to $Q^{14}$ in [KPS23]) via the refined topological
vertex [IKV09, IK17]. Alternatively, it can be read from the Nekrasov
partition function of rank 2 instantons, as explained in [KPS23, Section 3.3],
which also admits a combinatorial description. The refined GV/PT
correspondence is the following expression for the refined topological vertex
in terms of $\Omega_{d}$:

(39)
$Z^{\mathsf{ref}}_{\mathsf{PT}}=\mathrm{PE}\left(-\frac{q}{(1-qt)(1-q/t)}\sum_{d\geq
1}\Omega_{d}(q,t)Q^{d}\right)\,.$

This conjecture is an equivalent formulation of [KPS23, Conjecture 3.1]. The
specialization $t=1$ of the conjecture is precisely the (also conjectural)
unrefined GV/PT correspondence for $K_{{\mathbb{P}}^{2}}$. On the other hand,
the specialization $q=t$, known as the Nekrasov–Shatashvili limit (which
forgets the information of the perverse filtration, namely
$\Omega_{d}(q,q)=\overline{\mathit{IE}}(M_{d,\chi},q))$, was shown in [Bou23]
to be related to the relative Gromov–Witten theory of the pair
$({\mathbb{P}}^{2},E)$ where $E\subseteq{\mathbb{P}}^{2}$ is a smooth elliptic
curve.

######  Remark 6.10.

The numbers $N_{d}^{j_{L},j_{R}}$ are defined in [CKK14, (8.1)] from
$Z^{\mathsf{ref}}_{\mathsf{PT}}$; see Table 1 in loc. cit. for these numbers
up to $d=7$. Conjecture (39) is the statement that our definition (38) matches
with theirs.

######  Remark 6.11.

It is quite interesting to notice the similarity between (39) and the BPS
integrality formulas, for instance (34). Indeed, the $Q^{d}$ coefficient of

(40) $\mathrm{PE}\left(-\frac{q}{1-qt}\sum_{d\geq
1}\Omega_{d}(q,t)Q^{d}\right)$

is the generating function keeping track of the numbers
$\dim\mathrm{gr}^{P}_{i}H^{i+j}({\mathfrak{M}}_{d,0})$ where $P_{\bullet}$ is
a stacky perverse filtration as in [Dav23]. We do not have any good
explanation for the apparent similarity between (39) and (40).

Our calculation of the cohomology rings allow us to compute $\Omega_{d}$ for
$d\leq 5$, and thus check the refined GV/PT conjecture up to $d=5$.
Previously, this was known only up to $d=4$ [KPS23, Theorem 0.7], using the
calculation of the cohomology ring in [CM17].

######  Theorem 6.12.

The refined GV/PT correspondence (39) holds up to order $Q^{5}$.

###  6.3. Euler characteristics of line bundles.

We finish with a conjecture of Chung–Moon:

######  Conjecture 6.13 ([CM17, Conjecture 8.2]).

For $d\geq 1$ and coprime $\chi$, we have

(41) $\chi\big{(}M_{d,\chi},\,m\cdot c_{0}(2)\big{)}=\binom{m+3d-1}{m}.$

Here we write $c_{0}(2)$ also to denote its associated line bundle, i.e. the
pull-back of $\mathcal{O}(1)$ under the Hilbert–Chow map. The Euler
characteristics in (41) are known as generalized $K$-theoretic Donaldson
numbers, and are relevant to the strange duality conjecture [CM17, Section 8]
for $\mathbb{P}^{2}$. Note that this is formulated only for $M_{d,1}$ (and
verified for $d\leq 4$) in [CM17]. The next proposition justifies our
extension of the conjecture to arbitrary $\chi$ coprime to $d$.

######  Proposition 6.14.

For a fixed $d$, the Euler characteristic in (41) is independent of the choice
on $\chi$. In particular, Conjecture 6.13 holds for $M_{d,1}$ if and only if
it holds for all coprime $M_{d,\chi}$.

###### Proof.

We first observe that the Euler characteristics in question are computed by
the Hilbert polynomial of the derived push-forward
$Rh_{*}\mathcal{O}_{M_{d,\chi}}$, where

$h:M_{d,\chi}\longrightarrow|d\cdot H|$

is the Hilbert–Chow map. Thus it suffices to show that
$Rh_{*}\mathcal{O}_{M_{d,\chi}}$ is $\chi$-independent over the base $|d\cdot
H|={\mathbb{P}}^{b}$. This follows from [MS23, Theorem 0.4] combined with the
theory of Hodge modules. Indeed, for any $\chi$ coprime to $d$, let
$\pi:\mathcal{C}\to U$ be the universal smooth curve over
$U\subset\mathbb{P}^{b}$; recall that [MS23] we have an isomorphism

(42)
$Rh_{*}\mathbb{Q}_{M_{d,\chi}}[n]\simeq\bigoplus_{i=0}^{2g}\mathrm{IC}\left(\wedge^{i}R^{1}\pi_{*}\mathbb{Q}_{\mathcal{C}}\right)[-i+g]$

in the bounded derived category $D^{b}(\mathrm{MHM}({\mathbb{P}}^{b}))$, where
$n=\dim M_{d,\chi}$, and

$\mathbb{Q}_{M_{d,\chi}}[n]=({\mathcal{O}}_{M_{d,\chi}},F_{\bullet}),\quad
0=F_{-1}{\mathcal{O}}_{M_{d,\chi}}\subset
F_{0}{\mathcal{O}}_{M_{d,\chi}}={\mathcal{O}}_{M_{d,\chi}}$

is the trivial Hodge module. Consider its associated de Rham complex

$\mathrm{DR}(\mathcal{O}_{M_{d,\chi}})=\left[\mathcal{O}_{M_{d,\chi}}\to\Omega_{M_{d,\chi}}^{1}\to\cdots\to\Omega_{M_{d,\chi}}^{n}\right][n]$

with the induced Hodge filtration
$F_{\bullet}\mathrm{DR}(\mathcal{O}_{M_{d,\chi}})$. A direct computation
yields

$\mathrm{gr}^{F}_{-\ell}\mathrm{DR}(\mathcal{O}_{M_{d,\chi}})=\Omega_{M_{d,\chi}}^{\ell}[n-\ell].$

Taking $\ell=0$, we have

$\displaystyle Rh_{*}{\mathcal{O}}_{M_{d,\chi}}$
$\displaystyle=Rh_{*}\left(\mathrm{gr}^{F}_{0}\mathrm{DR}({\mathcal{O}}_{M_{d,\chi}})\right)=\mathrm{gr}^{F}_{0}\mathrm{DR}\left(Rh_{*}\mathbb{Q}_{M_{d,\chi}}[n]\right)$
$\displaystyle\simeq\mathrm{gr}^{F}_{0}\mathrm{DR}\left(\bigoplus_{i=0}^{2g}\mathrm{IC}\left(\wedge^{i}R^{1}\pi_{*}\mathbb{Q}_{\mathcal{C}}\right)[-i+g]\right)$

is independent of $\chi$, where we used in the second equality the
commutativity of $\mathrm{gr}_{\bullet}^{F}\mathrm{DR}(-)$ with derived push-
forward [Sai88, Section 2.3.7]. This finishes the proof. ∎

######  Theorem 6.15.

Conjecture 6.13 holds for all coprime moduli spaces $M_{5,\chi}$.

###### Proof.

By Proposition 6.14, it suffices to consider $M=M_{5,1}$. We first determine
the point class in $H^{*}(M,\mathbb{Q})$. Recall that $M_{d,\chi}$ has only
$(p,p)$-classes [Bou22, Theorem 0.4.1]. It follows that

(43) $\chi(M,\mathcal{O}_{M})=\sum_{k\geq 0}(-1)^{k}\dim
H^{k}(M,\mathcal{O}_{M})=\sum_{k\geq 0}(-1)^{k}h^{0,k}(M)=1.$

On the other hand, the Hirzebruch–Riemann–Roch theorem gives

(44)
$\chi(M,\mathcal{O}_{M})=\int_{M}\mathrm{ch}(\mathcal{O}_{M})\cdot\mathrm{td}(M)=\int_{M}\mathrm{td}(M).$

The Todd class $\mathrm{td}(M)$ can be written in terms of the normalized
tautological classes $c_{k}(j)$ using the expression of the tangent bundle

$\mathcal{T}_{M}=-Rq_{*}R\mathcal{H}\kern-1.0pt\mathit{om}(\mathbb{F},\mathbb{F})+\mathcal{O}_{M}\in
K(M),$

see for example [KPS23, Section 2.3.3]. Combining Equations (43) and (44)
gives the point class in $H^{*}(M,\mathbb{Q})$ in terms of the normalized
classes191919Explicitly, it is represented by
$\frac{7812500000}{62868347}c_{0}(2)^{8}c_{2}(2)^{6}$.. The proof of Theorem
6.15 is then a straightforward verification in the cohomology ring, using the
Hirzebruch–Riemann–Roch formula

$\chi\big{(}M,m\cdot c_{0}(2)\big{)}=\int_{M}\mathrm{ch}(m\cdot
c_{0}(2))\cdot\mathrm{td}(M).$

Note that the LHS of (41) is a polynomial in the variable $m\in\mathbb{Z}$, so
we only need to check the equality for a finite number of values. ∎

##  Appendix A An identity in the quadratic descendent algebra

In this appendix we state and prove an identity involving the interaction
between the Virasoro derivations ${\mathsf{R}}_{n}$ and a special class $C$ in
(the completion of) the quadratic descendent algebra
${\mathbb{D}}\otimes{\mathbb{D}}$. This is a purely formal identity and it
holds for the descendent algebra ${\mathbb{D}}={\mathbb{D}}^{X}$ of an
arbitrary target variety $X$ (in the main text $X$ is always
${\mathbb{P}}^{2}$); for simplicity, we will assume that $X$ does not have odd
cohomology and that $\dim(X)$ is even, but this can be adapted with minor
changes in the signs. This identity is used in the proof that the Virasoro
operators preserve the ideal of ${\mathsf{GMR}}$ and we hope that it can be of
use in other contexts in the study of Virasoro constraints.

Define $C$ by

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

where

$\sum_{i}\gamma_{i}^{L}\otimes\gamma_{i}^{R}=\Delta_{\ast}{\mathrm{td}}(X)$

is the Künneth decomposition of the push-forward of ${\mathrm{td}}(X)$ along
the diagonal and $\gamma_{i}^{L}\in H^{2d_{i}^{L}}(X)$. Let
$C_{j}\in{\mathbb{D}}\otimes{\mathbb{D}}$ be the degree $2j$ part of $C$.
Recall that $C_{j}$ is a natural lift to the quadratic descendent algebra of
the Chern class
$c_{j}(-R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}},{\mathcal{F}}^{\prime}))$.

We define also the operators

$\partial_{n}\colon{\mathbb{D}}\otimes{\mathbb{D}}\to{\mathbb{D}}\otimes{\mathbb{D}}$

by the formula

$\partial_{n}={\mathsf{R}}_{n}\otimes\textup{id}+\sum_{k=-1}^{n}\binom{n+1}{k+1}\textup{id}\otimes{\mathsf{R}}_{k}\,.$

Note that both the class $C$ and the operators $\partial_{n}$ make sense in
${\mathbb{D}}_{\alpha}\otimes{\mathbb{D}}_{\alpha^{\prime}}$ as well.

######  Theorem A.1.

For $n\geq 0$, we have

$\partial_{n}(C)=\left(\sum_{0\leq a+b\leq
n}\frac{a!(n-a)!}{(n-a-b)!}\sum_{i}(-1)^{d_{i}^{L}+1}{\mathrm{ch}}_{a}(\gamma_{i}^{L})\otimes{\mathrm{ch}}_{b}(\gamma_{i}^{R})\right)C$

in (the completion of)
${\mathbb{D}}_{\alpha}\otimes{\mathbb{D}}_{\alpha^{\prime}}$.

Note that for $n=0$, the formula reads (see for example (46) below)

$({\mathsf{R}}_{0}\otimes\textup{id}+\textup{id}\otimes{\mathsf{R}}_{0}+\textup{id}\otimes{\mathsf{R}}_{-1})(C)=-\chi(\alpha,\alpha^{\prime})C.$

We illustrate this special case first. Applying the derivation
${\mathsf{R}}_{0}\otimes\textup{id}+\textup{id}\otimes{\mathsf{R}}_{0}$ to the
definition of $C$ yields

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

which is just Newton’s identity for
$-R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}},{\mathcal{F}}^{\prime})$.
The $n=0$ case of Theorem A.1 then follows from the following lemma.

######  Lemma A.2.

We have

$(\textup{id}\otimes{\mathsf{R}}_{-1})(C_{j})=-({\mathsf{R}}_{-1}\otimes\textup{id})(C_{j})=-(j-1+\chi(\alpha,\alpha^{\prime}))C_{j-1}\,$

in ${\mathbb{D}}_{\alpha}\otimes{\mathbb{D}}_{\alpha^{\prime}}$.

###### Proof.

We have

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

where in the second line we replaced $b-1$ by $b$. Now the term in the sum
with $a=b=0$ is

(46)
$-\sum_{i}(-1)^{d_{i}^{L}}{\mathrm{ch}}_{0}(\gamma_{i}^{L})\otimes{\mathrm{ch}}_{0}(\gamma_{i}^{R})=-\sum_{i}(-1)^{d_{i}^{L}}\int_{X}{\mathrm{ch}}(\alpha)\cdot\gamma_{i}^{L}\int_{X}{\mathrm{ch}}(\alpha^{\prime})\cdot\gamma_{i}^{R}=-\chi(\alpha,\alpha^{\prime})\,.$

The terms with $(a,b)\neq(0,0)$ are, up to a minus sign, the ones in (45), so
we conclude the proof of the lemma for the
$\textup{id}\otimes{\mathsf{R}}_{-1}$. The statement for
${\mathsf{R}}_{-1}\otimes\textup{id}$ is similar; it can also be deduced from
the fact that
$-R{\mathcal{H}}\kern-1.0pt\mathit{om}_{p}({\mathcal{F}},{\mathcal{F}}^{\prime})$
is unaffected when we twist both ${\mathcal{F}}$ and ${\mathcal{F}}^{\prime}$
by a line bundle, which formally implies that its Chern classes are
annihilated by
${\mathsf{R}}_{-1}\otimes\textup{id}+\textup{id}\otimes{\mathsf{R}}_{-1}$. ∎

###### Proof of Theorem A.1.

Observe that $\partial_{n}$ is a derivation. When we apply $\partial_{n}$ to
$C$, we obtain a sum of the form

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

where $K_{a,b}$ are some constants. These constants are a sum of contributions
of each summand of $\partial_{n}$:

  1. (i)

When ${\mathsf{R}}_{n}\otimes\textup{id}$ hits
$(-1)^{b+d_{i}^{L}}(a+b-n-1)!{\mathrm{ch}}_{a-n}(\gamma_{i}^{L})\otimes{\mathrm{ch}}_{b}(\gamma_{i}^{R})$
we get

$(-1)^{b+d_{i}^{L}}(a+b-n-1)!(a)_{n+1}{\mathrm{ch}}_{a}(\gamma_{i}^{L})\otimes{\mathrm{ch}}_{b}(\gamma_{i}^{R})$

where $(a)_{m}$ denotes the falling factorial

$(a)_{m}=a(a-1)\ldots(a-m+1)\,.$

Thus, we get a contribution of $(a+b-n-1)!(a)_{n+1}$ to the constant $K_{a,b}$
coming from this summand. Note that if $a<n$ then $(a)_{n}=0$, so the formula
for the contribution makes sense as long as $a+b\geq n+1$. If $a+b\leq n$ then
the contribution of this term is also set to 0.

  2. (ii)

When $\textup{id}\otimes{\mathsf{R}}_{k}$ hits
$(-1)^{b-k+d_{i}^{L}}(a+b-k-1)!{\mathrm{ch}}_{a}(\gamma_{i}^{L})\otimes{\mathrm{ch}}_{b-k}(\gamma_{i}^{R})$
we get

$(-1)^{b-k+d_{i}^{L}}(a+b-k-1)!(b)_{k+1}{\mathrm{ch}}_{a}(\gamma_{i}^{L})\otimes{\mathrm{ch}}_{b}(\gamma_{i}^{R})\,.$

Hence, the summand $\binom{n+1}{k+1}\textup{id}\otimes{\mathsf{R}}_{k}$
contributes to $K_{a,b}$ with

$(-1)^{k}\binom{n+1}{k+1}(a+b-k-1)!(b)_{k+1}\,.$

As before, this formula holds as long as $a+b\geq k+1$.

Now for $(a,b)$ with $a+b>n$, we have

$K_{a,b}=(a+b-n-1)!(a)_{n+1}+\sum_{k=-1}^{n}(-1)^{k}\binom{n+1}{k+1}(a+b-k-1)!(b)_{k+1}=0\,,$

where the last equality results in an application of Lemma A.3 with
$m=n+1,\,\ell=k+1$, which we will prove next.

When $a+b\leq n$, we have

$\displaystyle K_{a,b}$
$\displaystyle=\sum_{k=-1}^{a+b-1}(-1)^{k}\binom{n+1}{k+1}(a+b-k-1)!(b)_{k+1}$
$\displaystyle=a!b!\sum_{k=-1}^{b-1}(-1)^{k}\binom{n+1}{k+1}\binom{a+b-k-1}{a}=-a!b!\binom{a+b-n-1}{b}$
$\displaystyle=(-1)^{b-1}\frac{a!(n-a)!}{(n-a-b)!}$

where the penultimate identity is (47). This finishes the proof. ∎

######  Lemma A.3.

Let $m\in{\mathbb{Z}}_{\geq 0}$ and $a,b\in{\mathbb{C}}$. Then the following
identity holds:

$(a)_{m}=\sum_{\ell=0}^{m}(-1)^{\ell}\binom{m}{\ell}(a+b-\ell)_{m-\ell}(b)_{\ell}\,.$

###### Proof.

This is a polynomial identity in $a$ and $b$, so it is enough to prove it for
$a,b\in{\mathbb{Z}}$ with $a,b>m$. In this case, we may replace the falling
factorials by a quotient of factorials, and the identity we want can be
rewritten as

(47)
$\binom{a+b-m}{b}=\sum_{\ell=0}^{m}(-1)^{\ell}\binom{m}{\ell}\binom{a+b-\ell}{a}\,.$

This follows by taking the $z^{a+b}$ coefficient in the identity of generating
series

$\frac{z^{a}}{(1-z)^{a+1-m}}=(1-z)^{m}\frac{z^{a}}{(1-z)^{a+1}}.\qed$

## References

  * [AB83] M. F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond., Ser. A, 308 (1983), pp. 523–615.
  * [Ach13] P. Achar, Equivariant mixed Hodge modules, available at https://www.math.lsu.edu/~pramod/docs/emhm.pdf, (2013).
  * [BBD82] A. A. Beilinson, J. Bernstein, and P. Deligne, Perverse sheaves. Astérisque 100, 172 p. (1982)., 1982.
  * [Bea86] A. Beauville, Sur l’anneau de Chow d’une variété abélienne, Math. Ann., 273 (1986), pp. 647–651.
  * [Bea95] A. Beauville, Sur la cohomologie de certains espaces de modules de fibrés vectoriels, in Geometry and analysis (Bombay, 1992), Tata Inst. Fund. Res., Bombay, 1995, pp. 37–40.
  * [BLM22] A. Bojko, W. Lim, and M. Moreira, Virasoro constraints for moduli of sheaves and vertex algebras, ArXiv:2210.05266, (2022), to appear in Invent. Math.
  * [BNR89] A. Beauville, M. S. Narasimhan, and S. Ramanan, Spectral curves and the generalized theta divisor, J. Reine Angew. Math., 398 (1989), pp. 169–179.
  * [Boj24] A. Bojko, Universal Virasoro constraints for quivers with relations, 2024\.
  * [Bou22] P. Bousseau, Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^{2}$, J. Algebr. Geom., 31 (2022), pp. 593–686.
  * [Bou23] P. Bousseau, A proof of N. Takahashi’s conjecture for $(\mathbb{P}^{2},E)$ and a refined sheaves/Gromov–Witten correspondence, Duke Math. J., 172 (2023), pp. 2895–2955.
  * [CC15] J. Choi and K. Chung, The geometry of the moduli space of one-dimensional sheaves, Sci. China, Math., 58 (2015), pp. 487–500.
  * [CC16] J. Choi and K. Chung, Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^{2}$, J. Math. Soc. Japan, 68 (2016), pp. 685–709.
  * [CKK14] J. Choi, S. Katz, and A. Klemm, The refined BPS index from stable pair invariants, Comm. Math. Phys., 328 (2014), pp. 903–954.
  * [CM17] K. Chung and H.-B. Moon, Chow ring of the moduli space of stable sheaves supported on quartic curves, Q. J. Math., 68 (2017), pp. 851–887.
  * [Dav22] B. Davison, BPS Lie algebras and the less perverse filtration on the preprojective CoHA, 2022\.
  * [Dav23] B. Davison, Nonabelian Hodge theory for stacks and a stacky $P=W$ conjecture, Adv. Math., 415 (2023), p. 61. Id/No 108889.
  * [dC16] M. A. A. de Cataldo, Perverse sheaves and the topology of algebraic varieties, 2016\.
  * [dCHM12] M. A. A. de Cataldo, T. Hausel, and L. Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case $A_{1}$, Ann. Math. (2), 175 (2012), pp. 1329–1407.
  * [dCM05] M. A. A. de Cataldo and L. Migliorini, The Hodge theory of algebraic maps, Ann. Sci. Éc. Norm. Supér. (4), 38 (2005), pp. 693–750.
  * [dCMS22] M. A. de Cataldo, D. Maulik, and J. Shen, Hitchin fibrations, abelian surfaces, and the $P=W$ conjecture, J. Am. Math. Soc., 35 (2022), pp. 911–953.
  * [DM91] C. Deninger and J. Murre, Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math., 422 (1991), pp. 201–219.
  * [DM11] J.-M. Drézet and M. Maican, On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics, Geom. Dedicata, 152 (2011), pp. 17–49.
  * [DM20] B. Davison and S. Meinhardt, Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras, Invent. Math., 221 (2020), pp. 777–871.
  * [EHX97] T. Eguchi, K. Hori, and C.-S. Xiong, Quantum cohomology and Virasoro algebra, Phys. Lett. B, 402 (1997), pp. 71–80.
  * [EK04] R. Earl and F. Kirwan, Complete sets of relations in the cohomology rings of moduli spaces of holomorphic bundles and parabolic bundles over a Riemann surface, Proc. Lond. Math. Soc. (3), 89 (2004), pp. 570–622.
  * [ES93] G. Ellingsrud and S. A. Strømme, Towards the Chow ring of the Hilbert scheme of $\mathbb{P}^{2}$, J. Reine Angew. Math., 441 (1993), pp. 33–44. Ann., 286 (1990), pp. 193–207.
  * [GS] D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at http://www2.macaulay2.com.
  * [HL10] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Cambridge: Cambridge University Press, 2nd ed. ed., 2010.
  * [HMMS22] T. Hausel, A. Mellit, A. Minets, and O. Schiffmann, $P=W$ via $\mathcal{H}_{2}$, ArXiv:2209.05429, (2022).
  * [Huy06] D. Huybrechts, Fourier–Mukai transforms in algebraic geometry, Oxford Math. Monogr., Oxford: Claren- don Press, 2006.
  * [IK17] A. Iqbal and C. Kozçaz, Refined topological strings on local $\mathbb{P}^{2}$, J. High Energy Phys., (2017), pp. 069, front matter+20.
  * [IKV09] A. Iqbal, C. Kozçaz, and C. Vafa, The refined topological vertex, J. High Energy Phys., (2009), pp. 069, 58\.
  * [Joy18] D. Joyce, Ringel–Hall style Lie algebra structures on the homology of moduli spaces. Preprint, 2018.
  * [Kir92] F. Kirwan, The cohomology rings of moduli spaces of bundles over Riemann surfaces, J. Am. Math. Soc., 5 (1992), pp. 853–906.
  * [Kon92] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., 147 (1992), pp. 1–23.
  * [KPS23] Y. Kononov, W. Pi, and J. Shen, Perverse filtrations, Chern filtrations, and refined BPS invariants for local $\mathbb{P}^{2}$, Adv. Math., 433 (2023), pp. Paper No. 109294, 29.
  * [KS11] M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, 2011\.
  * [Lin18] Y. Lin, Moduli spaces of stable pairs, Pacific J. Math., 294 (2018), pp. 123–158.
  * [LM24] W. Lim, M. Moreira, Virasoro constraints and representations for quiver moduli spaces, in preparation.
  * [LMP23] W. Lim, M. Moreira, and W. Pi, Cohomological $\chi$-dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb{P}^{2}$, ArXiv:2305.14193, (2023), to appear in Forum Math. Sigma.
  * [LP93a] J. Le Potier, Faisceaux semi-stables de dimension 1 sur le plan projectif, Rev. Roumaine Math. Pures Appl., 38 (1993), pp. 635–678.
  * [LP93b] J. Le Potier, Systèmes cohérents et structures de niveau, Astérisque, (1993), p. 143.
  * [Mai10] M. Maican, A duality result for moduli spaces of semistable sheaves supported on projective curves, Rend. Semin. Mat. Univ. Padova, 123 (2010), pp. 55–68.
  * [Mai13] M. Maican, The homology groups of certain moduli spaces of plane sheaves, Int. J. Math., 24 (2013), p. 42. Id/No 1350098.
  * [Mar07] E. Markman, Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces, Adv. Math., 208 (2007), pp. 622–646.
  * [MOOP22] M. Moreira, A. Oblomkov, A. Okounkov, and R. Pandharipande, Virasoro constraints for stable pairs on toric threefolds, Forum Math. Pi, 10 (2022), pp. Paper No. e20, 62.
  * [Mor22] M. Moreira, Virasoro conjecture for the stable pairs descendent theory of simply connected 3-folds (with applications to the Hilbert scheme of points of a surface), J. Lond. Math. Soc. (2), 106 (2022), pp. 154–191.
  * [MR15] S. Mozgovoy and M. Reineke, Intersection cohomology of moduli spaces of vector bundles over curves, 2015\.
  * [MS22] D. Maulik and J. Shen, The $P=W$ conjecture for $\mathrm{GL}_{n}$. arXiv:2209.02568, 2022.
  * [MS23] D. Maulik and J. Shen, Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol., 27 (2023), pp. 1539–1586.
  * [MSS11] L. Maxim, M. Saito, and J. Schürmann, Symmetric products of mixed Hodge modules, J. Math. Pures Appl. (9), 96 (2011), pp. 462–483.
  * [MSY23] D. Maulik, J. Shen, and Q. Yin, Perverse filtrations and fourier transforms, ArXiv:2308.13160, (2023).
  * [MT18] D. Maulik and Y. Toda, Gopakumar–Vafa invariants via vanishing cycles, Invent. Math., 213 (2018), pp. 1017–1097.
  * [MY14] D. Maulik and Z. Yun, Macdonald formula for curves with planar singularities, J. Reine Angew. Math., 694 (2014), pp. 27–48.
  * [NO16] N. Nekrasov and A. Okounkov, Membranes and sheaves, Algebr. Geom., 3 (2016), pp. 320–369.
  * [OY17] A. Oblomkov and Z. Yun, The cohomology ring of certain compactified Jacobians, 2017\.
  * [PS23] W. Pi and J. Shen, Generators for the cohomology ring of the moduli of 1-dimensional sheaves on $\mathbb{P}^{2}$, Algebr. Geom., 10 (2023), pp. 504–520.
  * [PT14] R. Pandharipande and R. P. Thomas, 13/2 ways of counting curves, in Moduli spaces, vol. 411 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2014, pp. 282–333.
  * [Sai88] M. Saito, Modules de Hodge polarisables., Publ. Res. Inst. Math. Sci., 24 (1988), pp. 849–995.
  * [She16] J. Shen, Cobordism invariants of the moduli space of stable pairs, J. Lond. Math. Soc., II. Ser., 94 (2016), pp. 427–446.
  * [Sim94] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math., (1994), pp. 47–129.
  * [SY22] J. Shen and Q. Yin, Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds, Duke Math. J., 171 (2022), pp. 209–241.
  * [vB23] D. van Bree, Virasoro constraints for moduli spaces of sheaves on surfaces, Forum Math. Sigma, 11 (2023), pp. Paper No. e4, 35.
  * [Wit91] E. Witten, Two-dimensional gravity and intersection theory on moduli space, in Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310.
  * [Yua14] Y. Yuan, Moduli spaces of semistable sheaves of dimension one on $\mathbb{P}^{2}$, Pure Appl. Math. Q., 10 (2014), pp. 723–766.
  * [Yua23a] Y. Yuan, On the perverse filtration of the moduli spaces of 1-dimensional sheaves on $\mathbb{P}^{2}$ and $P=C$ conjecture. 2023\.
  * [Yua24] Y. Yuan, Some Betti numbers of the moduli of 1-dimensional sheaves on $\mathbb{P}^{2}$, Forum Math., 36 (2024), pp. 215–244.