Virasoro constraints for moduli spaces of sheaves on surfaces

2.1 Eliminating fineness

The cohomology classes of (3) were constructed using a universal family of semistable sheaves on $X \times M$ . This universal family is not needed since we have always access to a twisted universal family $\mathcal {E}$ [Reference Căldăraru2]. Denote its Brauer class by $\alpha $ . Then $\mathcal {E}^{\otimes r}$ and $\det \mathcal {E}$ both have Brauer class $r\alpha $ , where r is the rank of $\mathcal {E}$ . In particular, $\mathcal {E}^{\otimes r} \otimes ( \det {\mathcal {E}})^{-1}$ has Brauer class 0, that is, it is an ordinary sheaf. Therefore, we might compute the cohomology classes above by taking the Chern classes of this sheaf and then taking the r-th root on the level of cohomology. Finally, we note that this is independent of the twisted family chosen. Indeed, if $\mathcal {E}' = \mathcal {E} \otimes L$ is another family, for L a line bundle, then $\mathcal {E}^{\prime \otimes r} = L^{\otimes r} \otimes \mathcal {E}^ {\otimes r}$ and $\det \mathcal {E}' = L^{\otimes r} \otimes \det \mathcal {E}$ , hence $\mathcal {E}^{\prime \otimes r} \otimes \det (\mathcal {E}')^{-1} \cong \mathcal {E}^{\otimes r} \otimes (\det \mathcal {E})^{-1}$ .

2.2 The conjecture for small k

For small k, it is actually possible to verify the conjecture by providing identities for $\operatorname {\mathrm {ch}}_0(\gamma )$ and $\operatorname {\mathrm {ch}}_1(\gamma )$ . These equations are similar to Proposition 1 of Moreira [Reference Moreira20].

Proof. Let $I = H^{\geq 4}(X \times M, \mathbb {Q})$ . We can write $\operatorname {\mathrm {ch}}(\mathcal {E})$ as $r + c_1(\mathcal {E}) \ \mod I$ . Similarly, we can write $\operatorname {\mathrm {ch}}(\det (\mathcal {E})^ {-1/r})$ as $e^ {-c_1(\mathcal {E})/r} = 1 - c_1(\mathcal {E})/r \ \mod I$ . Their product is $r \ \mod I$ . We obtain that $\operatorname {\mathrm {ch}}_k(-\mathcal {E} \otimes \det \mathcal {E}^{-1/r})$ is $-r$ for $k = 0$ and $0$ for $k = 1$ . This second identity implies $\operatorname {\mathrm {ch}}_1(\gamma ) = 0$ . The first identity can be used to rewrite $\operatorname {\mathrm {ch}}_0(\gamma )$ as an integral:

$$ \begin{align*} \operatorname{\mathrm{ch}}_0(\gamma) = \pi_{M, *}(\pi_X^ *\gamma \cdot -r) = -r \pi_{M, *}\pi_X^*\gamma = -r \cdot\int_X\gamma \in H^0(M,\mathbb{Q}). \end{align*} $$

One minor annoyance is that the algebra $\mathbb {D}^X$ contains elements of negative degree, for example, $\operatorname {\mathrm {ch}}_0(1)$ has degree $-2$ . We will deal with these elements in the next lemma, telling us that we can essentially ignore these.

Proof. We verify this for $R_k$ , $T_k$ and $S_k$ separately. For $T_k$ it is immediate. For $R_k$ , we note that $R_k(\operatorname {\mathrm {ch}}_i(\gamma )D) = R_k(\operatorname {\mathrm {ch}}_i(\gamma ))D + \operatorname {\mathrm {ch}}_i(\gamma )R_kD$ . But note that $R_k(\operatorname {\mathrm {ch}}_i(\gamma )) = 0$ because either this has negative degree or there is a zero in the product in the definition of $R_k$ . For $S_k$ , we use again that $R_{-1}$ is a derivation to see that

$$ \begin{align*} S_k(\operatorname{\mathrm{ch}}_i(\gamma)D) = \frac{(k+1)!}{r} (\operatorname{\mathrm{ch}}_{k+1}(\mathbf{p})DR_{-1}\operatorname{\mathrm{ch}}_i(\gamma) + \operatorname{\mathrm{ch}}_{i}(\gamma)R_{-1}(\operatorname{\mathrm{ch}}_{k+1}(\mathbf{p})D)). \end{align*} $$

But $R_{-1}(\operatorname {\mathrm {ch}}_i(\gamma ))$ has strictly negative degree, so it vanishes in the cohomology of M.

Proof. Lemma 2.1 tells us that $\operatorname {\mathrm {ch}}_1(\mathbf {p}) = 0$ in $H^\bullet (M, \mathbb {Q})$ , hence $T_k(\operatorname {\mathrm {ch}}_1(\mathbf {p})) = 0$ . By the same lemma,

$$ \begin{align*} R_k(\operatorname{\mathrm{ch}}_1(\mathbf{p})D) = R_k(\operatorname{\mathrm{ch}}_1(\mathbf{p}))D = (k+1)!\operatorname{\mathrm{ch}}_{k+1}(\mathbf{p})D \end{align*} $$

and

$$ \begin{align*} S_k(\operatorname{\mathrm{ch}}_1(\mathbf{p})D) = \frac{(k+1)!}{r}R_{-1}(\operatorname{\mathrm{ch}}_{k+1}(\mathbf{p})\operatorname{\mathrm{ch}}_1(\mathbf{p})D) = \frac{(k+1)!}{r}\operatorname{\mathrm{ch}}_{k+1}(\mathbf{p})\operatorname{\mathrm{ch}}_0(\mathbf{p})D. \end{align*} $$

Now, $\operatorname {\mathrm {ch}}_0(\mathbf {p}) = -r$ by the other identity from Lemma 2.1. So we are done.

Proof. For $k = -1$ , note that

$$ \begin{align*} S_{-1}D = \frac1rR_{-1}(\operatorname{\mathrm{ch}}_0(\mathbf{p})D) = \frac1r \operatorname{\mathrm{ch}}_{-1}(\mathbf{p})D + \frac1r \operatorname{\mathrm{ch}}_0(\mathbf{p}) R_{-1}D. \end{align*} $$

In $H^\bullet (M, \mathbb {Q})$ , we have that $\operatorname {\mathrm {ch}}_{-1}(\mathbf {p}) = 0$ for degree reasons and $\operatorname {\mathrm {ch}}_0(\mathbf {p}) = -r$ by Lemma 2.1. Hence, we get $S_{-1} = - R_{-1}$ . Next, we show $T_{-1} = 0$ . The second sum in equation (2) is empty. In the first sum, to have both $a + d^L - 2$ and $b + d^R - 2$ nonnegative, we must have $a + b - 2 = a + b + d^L - 2 + d^R - 2 \geq 0$ , but $a + b = 1$ .

For $k = 0$ , again we assume that $D = \prod _j \operatorname {\mathrm {ch}}_{i_j}(\gamma _j)$ with $\gamma _j$ of pure degree. By induction, we see that $R_0D = (\deg D) D$ . By using that $\operatorname {\mathrm {ch}}_0(\mathbf {p}) = -r$ and $\operatorname {\mathrm {ch}}_1(\mathbf {p}) = 0$ , we compute that $S_{0}D = -D$ . Finally, consider $T_0$ . In the second sum, we need to consider the Künneth decomposition of $\mathbf {p}$ , which is $\mathbf {p} \otimes \mathbf {p}$ . Also, since X only has $(p,p)$ -cohomology, $\chi (X, \mathcal {O}_X) = 1$ . Since $a + b = 0$ in this sum, $a = b = 0$ , and the second sum becomes $\operatorname {\mathrm {ch}}_0(\mathbf {p})\operatorname {\mathrm {ch}}_0(\mathbf {p}) = r^2$ .

In the first sum, we have that $a + b = 2$ . Since $\operatorname {\mathrm {ch}}_1(\gamma ) = 0$ by Lemma 2.1, we do not need to consider $a = b = 1$ . If $a = 0$ and $b = 2$ , we must have $d^L = 2$ and $d^R = 0$ ; otherwise the factorials become negative. So we only have to deal with the Künneth component in $H^4(X, \mathbb {Q}) \otimes H^0(X, \mathbb {Q})$ , which is $\mathbf {p} \otimes 1$ . So for $a = 0$ and $b = 2$ , we get $-\operatorname {\mathrm {ch}}_0(\mathbf {p})\operatorname {\mathrm {ch}}_2(1) = r\operatorname {\mathrm {ch}}_2(1)$ . For $a = 2$ and $b = 0$ , we get the same result, so taking everything together we find that $T_0 = -2r\operatorname {\mathrm {ch}}_2(1) + r^2$ . Since $\operatorname {\mathrm {ch}}_2(1)$ is of degree zero, we can compute it by picking a point $[E] \in M$ and noticing that $\operatorname {\mathrm {ch}}_2(1)|_{[E]} = \operatorname {\mathrm {ch}}_2(-E \otimes \det E^{-1/r})$ by the push-pull formula. We have fixed r, $\det E$ and $c_2$ , so we can calculate this in a similar manner to Lemma 2.1 and obtain that

$$ \begin{align*} 2r\operatorname{\mathrm{ch}}_2(1) = 2r\operatorname{\mathrm{ch}}_2(-E\otimes \det E^{-1/r}) = 2r\frac{2rc_2 - c_1(\Delta)^2(r - 1)}{2r} = \operatorname{\mathrm{vdim}} M + (r^2 - 1)\chi(X, \mathcal{O}_X). \end{align*} $$

Keeping in mind that $\chi (X, \mathcal {O}_X) = 1$ , we find that $T_0 = -\operatorname {\mathrm {vdim}} M - r^2 + 1 + r^2 = -\operatorname {\mathrm {vdim}} M + 1$ . Finally, $T_0 + S_0 = -\operatorname {\mathrm {vdim}} M$ . Then we obtain

$$ \begin{align*} \int_{[M]^{\mathrm{vir}}} \mathcal{L}_0D = (\deg D - \operatorname{\mathrm{vdim}} M) \int_{[M]^{\mathrm{vir}}} D. \end{align*} $$

If the integral is nonzero, then $\deg D = \operatorname {\mathrm {vdim}} M$ , in which case the first factor vanishes.

2.3 The Virasoro bracket

The operators $L^{\text {GW}}_k$ from Gromov–Witten theory satisfy the Virasoro bracket. In our situation, there is a Virasoro bracket as well, but it requires new notation. This notation is much more convenient than the notation we employed before, which we only use because of its history and because the new notation was only discovered after the rest of the paper was already written.

Note that $h_i(\gamma )$ always has degree i. The next proposition is immediate.

Definition 2.8. Fix integers r and k with $k \geq -1$ . Define the operator $R_k^+$ on $\mathbb {D}_+^X$ as a derivation, which acts as $R_k^+(h_i(\gamma )) = ih_{i+k}(\gamma )$ on generators. For $\gamma _1$ and $\gamma _2$ of pure degree, define

$$ \begin{align*} t_k(\gamma_1, \gamma_2) = \sum_{a + b = k} (-1)^{2 - \deg \gamma_1}h_a(\gamma_1)h_b(\gamma_2). \end{align*} $$

Extend the definition by bilinearity. Then define the operator $T_k^+$ as multiplication by the constant element

$$ \begin{align*} T_k^+ = \sum_i t_k(\gamma_i^L, \gamma_i^R), \end{align*} $$

where $\sum _i \gamma _i^L \otimes \gamma _i^R = \Delta _*\operatorname {\mathrm {td}}_X$ , the Künneth decomposition of the Todd class of X. Finally, let $S_k^+$ be defined by $S_k^+D = \frac {1}{r}R_{-1}(h_{k+1}(\mathbf {p})D)$ . Let $L_k^+ = R_k^+ + T_k^+$ and $\mathcal {L}_k^+ = L_k^+ + S_k$ .

Note the strange sign convention in the definition of $t_k$ . One should read $2 = \dim X$ here so that we have a natural candidate for a generalisation for curves or higher-dimensional X. These signs agree with the description of the Virasoro constraints in PT-theory for threefolds; see [Reference Moreira20].

The operators $R_k^+$ , $T_k^+$ and $S_k^+$ almost agree with their counterparts of Definition 1.2. In fact for $k \geq 0$ they agree on all elements of $\mathbb {D}^X_+$ ; see below. For $k = -1$ , we have $T_k^+ = T_k = 0$ , so they are the same as well. Finally, $R_{-1}^+$ and $R_{-1}$ agree on elements of positive degree but not on elements of degree zero. Indeed, $R_{-1}$ sends elements of degree zero to elements of degree $-1$ , while $R_{-1}^+$ simply sends those to zero.

Proposition 2.9. For $k \geq 0$ , we have $R_k^+ = R_k$ , $T_k^+ = T_k$ and $S_k^+ = S_k$ . Furthermore, Conjecture 1.4 holds if and only if for every $k \geq -1$ , $r \geq 1$ , $\Delta $ a line bundle on X, $c_2$ an integer and H a polarisation on X such that $M=M_X^H(r, \Delta , c_2)$ contains only stable sheaves, and D any element of $\mathbb {D}_+^ X$ , we have

(4) $$ \begin{align} \int_{[M]^{\text{vir}}} \mathcal{L}^+_kD = 0. \end{align} $$

Another advantage of using the new notation is that $L_k^+ = R_k^+ + T_k^+$ satisfies the Virasoro bracket in full generality. This is not true if we consider only $R_k + T_k$ on $\mathbb {D}^X$ . It is also not true in the stable pair setting [Reference Moreira, Oblomkov, Okounkov and Pandharipande21], where the bracket is only satisfied after introducing a new formal symbol and using a weaker notion of equality of operators. Finally, note that $L_k^+$ does not depend on the rank r, only $\mathcal {L}_k^+$ does so.

Proposition 2.10. The operator $L^+_k$ satisfies the Virasoro bracket as operators on $\mathbb {D}_+^X$ , that is,

(5) $$ \begin{align} [L^+_k, L^+_m] = (m - k)L^+_{m + k} \end{align} $$

for all $m, k \geq -1$ .

For the proof, we first prove two lemmas.

Proof. The commutator $[R^+_k, R^+_m]$ is again a derivation, and we have

$$ \begin{align*} R^+_kR^+_mh_i(\gamma) - R^+_mR^+_kh_i(\gamma) = i(i+m)h_{i + m +k}(\gamma) - i(i + k)h_{i+m+k}(\gamma) = (m - k)R^+_{m+k}h_{i}(\gamma). \end{align*} $$

So they also agree on generators.

Lemma 2.12. For all $m, k \geq -1$ , equation (5) is equivalent to

(6) $$ \begin{align} R^+_k(T^+_m) - R^+_m(T^+_k) = (m - k)T^+_{m+k}. \end{align} $$

Proof. Expanding $L^+_k = R^+_k + T^+_k$ in equation (5) gives the equation

$$ \begin{align*} [R^+_k, R^+_m] + [R^+_k, T^+_m] + [T^+_k, R^+_m] + [T^+_k, T^+_m] = (m - k)R^+_{m+k} + (m - k)T^+_{m+k}. \end{align*} $$

By Lemma 2.11 and by noting that $[T^+_k, T^+_m] = 0$ , we get

(7) $$ \begin{align} [R^+_k, T^+_m] + [T^+_k, R^+_m] = (m - k)T^+_{m+k}. \end{align} $$

Finally, note that $[R^+_k, T^+_m] = R^+_k(T^+_m)$ since $R^+_k$ is a derivation and $T^+_m$ is a constant as

$$ \begin{align*} R^+_k(T^+_mD) - T^+_mR^+_k(D) = R^+_k(T^+_m)D + T^+_mR^+_k(D) - T^+_mR^+_k(D) = R^+_k(T^+_m)D \end{align*} $$

holds for all D.

Proof of Proposition 2.10

We will show that the following analog of equation (6) holds for $\gamma _1$ and $\gamma _2$ of pure degree:

$$ \begin{align*} R_k^+(t_m(\gamma_1, \gamma_2)) - R^+_m(t_k(\gamma_1, \gamma_2)) = (m - k)t_{m+k}(\gamma_1, \gamma_2). \end{align*} $$

Given the above expression of $T_k^+$ , this immediately implies equation (6) and hence completes the proof of the proposition. Assume for simplicity that $(-1)^{2 - \deg \gamma } = 1$ , this does not affect the proof in an essential way. Note that both sides of the equation are a linear combination of $h_a(\gamma _1)h_b(\gamma _2)$ with $a + b = m + k$ . We count how often each $h_a(\gamma _1)h_b(\gamma _2)$ occurs on the left-hand side.

We can get terms $h_a(\gamma _1)h_b(\gamma _2)$ by either applying $R_k$ to $h_{a - k}(\gamma _1)h_b(\gamma _2)$ or $h_{a}(\gamma _1)h_{b - k}(\gamma _2)$ or by applying $R_m$ to $h_{a - m}(\gamma _1)h_b(\gamma _2)$ or $h_a(\gamma _1)h_{b - m}(\gamma _2)$ . But if $b - m < 0$ , for example, we get zero automatically for $R_m(h_a(\gamma _1)h_{b - m}(\gamma _2))$ . Therefore, it is useful to distinguish whether a is less than, equal to or more than k and similarly for b. For example, if $a> k$ and $b> k$ , then we also have $a < m$ and $b < m$ . So in this case, only $R_k(h_{a-k}(\gamma _1)h_b(\gamma _2))$ and $R_k(h_{a}(\gamma _1)h_{b - k}(\gamma _2))$ will contribute. The first contributes $(a - k)h_a(\gamma _1)h_b(\gamma _2)$ , and the second contributes $(b - k)h_a(\gamma _1)h_b(\gamma _2)$ . So the total contribution is $a - k + b - k = m + k - 2k = m - k$ , which is exactly the same as the coefficient on the right-hand side. The other cases are similar.

Finally, we list some more bracket relations. First, we have

$$ \begin{align*} [L_n^+, h_k(\mathbf{p})] = kh_{n+k}(\mathbf{p}). \end{align*} $$

This relation also appears in [Reference Moreira, Oblomkov, Okounkov and Pandharipande21]. We also have the relation

$$ \begin{align*} [L_{-1}^+, S_k^+] = (k+1)S_{k-1}^+. \end{align*} $$

This implies that $[L_{-1}^+, \mathcal {L}_k^+] = (k+1)\mathcal {L}_{k-1}$ . In view of the above remark, one might hope that a similar inductive argument might be used to shed light on some parts of Conjecture 1.4, but the author has not succeeded in this.

2.4 Deformation invariance

Let S be a smooth $\mathbb {C}$ -scheme and consider a smooth family of surfaces $\mathcal {X} \to S$ . Let r be a number, $\Delta $ a line bundle on $\mathcal {X}$ and $c_2$ be a cohomology class in $H^4(\mathcal {X}, \mathbb {Z})$ . Then for each $s \in S$ , we can construct the moduli space $M_s$ of stable sheaves on $\mathcal {X}_s$ of rank r, determinant $\Delta |_{\mathcal {X}_s}$ and second Chern class $c_2|_{\mathcal {X}_s}$ .

3.1 Generalities on smooth projective toric surfaces

We briefly recall the basic theory of toric varieties that we need. This material can be found in [Reference Fulton7]. Assume that X is a toric surface. Associated to X is a fan $\Delta $ in N, a free abelian group of rank two. Let $M = N^\vee $ . We then have the natural pairing $\langle -,-\rangle : M \otimes N \to \mathbb {Z}$ . Associated to each cone $\sigma \in \Delta $ is the set $S_\sigma = \{ m \in M \mid \langle m, s \rangle \geq 0 $ for all $s \in \sigma \}$ . Denote by $\mathbb {C}[S_\sigma ]$ the ring generated by formal symbols $z^m$ for $m \in S_\sigma $ with multiplication $z^{m_1} \cdot z^{m_2} = z^{m_1 + m_2}$ , and let $U_{\sigma }$ be $\operatorname {\mathrm {Spec}} \mathbb {C}[S_\sigma ]$ . If $\sigma _1 \subseteq \sigma _2$ is an inclusion of cones, then we have a canonical open embedding $U_{\sigma _1} \subseteq U_{\sigma _2}$ . If we glue the $U_{\sigma }$ along all possible inclusions, we recover X. Recall that the fact that X is proper is equivalent to the union of the cones in $\Delta $ being equal to N. If this is the case, note the following: Each two-dimensional cone $\sigma $ is bordered by two rays $\rho _1$ and $\rho _2$ . Then X is smooth if and only if the primitive generators $v_1$ and $v_2$ of $\rho _1$ resp. $\rho _2$ form a basis of N, and this holds for each two-dimensional cone $\sigma $ . Finally, recall that any smooth proper surface is projective [Reference Liu18, Sec. 9.3.1].

The Chow ring of a smooth toric variety can be computed as follows. Enumerate the rays in $\Delta $ as $\rho _1, \rho _2, \ldots ,\rho _d$ with primitive generators $v_1, v_2, \ldots , v_d$ . For each $1 \leq i \leq d$ , we have a generator $D_i$ . These are subject to the following relations:

1. 1. For each $m \in M$ , add the relation $\sum _{i = 1}^d \langle m, v_i\rangle D_i = 0$ .

2. 2. For each subset A of $\{1, \ldots , d\}$ such that the $v_i$ for $i \in A$ do not generate a cone of $\Delta $ add the relation $\prod _{i \in A} D_i = 0$ .

For the first type of relation, one can restrict to a basis of M. In this language, the class of the canonical sheaf $\omega _X$ is $- \sum _{i = 1}^d D_i$ . In particular, $-\omega _X$ is effective.

3.2 Equivariant sheaves

The trivial cone $\{0\}$ in $\Delta $ corresponds to a two-dimensional torus $T = \operatorname {\mathrm {Spec}} \mathbb {C}[U_{\{0\}}] = \operatorname {\mathrm {Spec}} \mathbb {C}[M]$ . Then T acts on itself via left multiplication, and this action can always be uniquely extended to X. The $U_\sigma $ are preserved under the action. We are interested in coherent sheaves which are equivariant under this action. We recall the definition.

Definition 3.2. Let G be a group scheme and X a G-scheme (e.g., X is toric and G is the corresponding torus). Denote by $\mu : G \times G \to G$ the multiplication and by $\sigma : G \times X \to X$ the action. An equivariant structure on a sheaf F is an isomorphism

$$ \begin{align*} \phi : \sigma^* F \to \pi_X^* F \end{align*} $$

of sheaves on $G \times X$ such that the cocycle condition holds on $G\times G\times X$ : $\pi _{23}^*\phi \circ (\operatorname {\mathrm {id}}_G \times \sigma )^* \phi = (\mu \times \operatorname {\mathrm {id}}_X)^*\phi $ , where $\pi _{23}$ is the projection to the second and third factor.

If X and G are affine, then an equivariant sheaf F is just a module over $\Gamma (X, \mathcal {O}_X)$ together with a coaction of the coalgebra $\Gamma (G, \mathcal {O}_G)$ . In particular, if X is a toric variety and $G = T$ , the corresponding torus, we can describe an equivariant sheaf by specifying for each $\sigma $ a module over $\Gamma (U_{\sigma }, \mathcal {O}_X)$ with a $\Gamma (T, \mathcal {O}_T)$ -action such that for $\sigma _1$ and $\sigma _2$ , the modules and their actions agree on the overlap $U_{\sigma _1 \cap \sigma _2}$ . Clearly, if a sheaf has an equivariant structure, then it has several by multiplying with a character of T.

We can exploit the affine cover $U_{\sigma }$ of X to find a combinatorial description of equivariant sheaves. We will use this description to find all equivariant sheaves with certain numerical invariants. We introduce the following combinatorial data due to Perling [Reference Perling30]:

An equivariant sheaf F on an affine toric variety $U_\sigma $ gives rise to a $\sigma $ -family $\hat {F}$ as follows. First, we identify F with the $\mathbb {C}[S_\sigma ]$ -module $H^0(U_\sigma , F)$ . Then the affine group $T = \operatorname {\mathrm {Spec}} \mathbb {C}[M]$ acts on F, and since every action of a torus is diagonisable, F decomposes into weight spaces as $F = \bigoplus _{m \in M} \hat {F}_m$ . Multiplication by $z^s \in \mathbb {C}[S_{\sigma }]$ induces a map $\hat {F}_m \to \hat {F}_{m + s}$ . It is not difficult to see that this assignment extends to an equivalence of categories between equivariant coherent sheaves on $U_\sigma $ and finite $\sigma $ -families. In the following, the notation $\hat {F}$ will always mean the $\sigma $ -family associated to a coherent sheaf F.

Let $\sigma $ be a two-dimensional cone. For the smooth affine $U_\sigma $ ; there is a more concrete description of $S_\sigma $ , and hence of the $\sigma $ -families. Let $v_1$ and $v_2$ generate the two boundary rays of $\sigma $ , then by smoothness this is a basis of N. Hence, we obtain a dual basis $w_1$ , $w_2$ for M and $S_\sigma $ is exactly the set of positive linear combinations of the $w_i$ . This implies that $\mathbb {C}[S_\sigma ] \cong \mathbb {C}[z^{w_1}, z^{w_2}]$ , the usual polynomial ring in two variables. Let $\hat {F}$ be a $\sigma $ -family. Define $\hat {F}(n_1, n_2)$ as $\hat {F}_{n_1w_1 + n_2w_2}$ for integers $n_1, n_2$ . Note we have maps

(8) $$ \begin{align} \hat{F}(n_1, n_2) \to \hat{F}(n_1 + 1, n_2) \quad\text{and}\quad \hat{F}(n_1, n_2) \to \hat{F}(n_1, n_2 + 1) \end{align} $$

by multiplication with $z^{w_1}$ and $z^{w_2}$ , respectively. The $\hat {F}(n_1, n_2)$ together with these two maps completely determine $\hat {F}$ . In fact, we obtain again an equivalence of categories between $\sigma $ -families $\hat {F}$ and families $\hat {F}(n_1, n_2)$ with maps as in equation (8) which make all the squares commute. It is convenient to picture a lattice $\mathbb {Z}^2$ with $\hat {F}(n_1, n_2)$ sitting at the point $(n_1, n_2)$ , with horizontal maps going to the right and vertical maps going upwards.

We will now explain how these $\sigma $ -families glue to produce equivariant coherent sheaves on X. This is easier to describe in the case of a torsion-free sheaf. Since this is the only case we will need, we assume that all our equivariant sheaves are torsion-free from now on. We have the following characterisation in terms of $\sigma $ -families.

The $\sigma $ -family $\hat {F}$ of an equivariant coherent sheaf F on $U_\sigma $ is finitely generated. Hence, for large $n_1$ , $n_2$ , the inclusions of equation (8) are actually identities. Assume without loss of generality that this limiting space is $\mathbb {C}^r$ . Then r is the rank of F. Furthermore, for each fixed $n_1$ , we can produce a filtration of $\mathbb {C}^n$ . Indeed, since the $\sigma $ -family is finite, the space $\hat {F}(n_1, n_2)$ is constant for sufficiently large $n_2$ . Denote this space by $\hat {F}(n_1, \infty )$ . Varying $n_1$ gives us sequence of inclusions

(9) $$ \begin{align} \ldots \subseteq \hat{F}(n - 1, \infty) \subseteq \hat{F}(n, \infty), \subseteq \hat{F}+1,\infty) \subseteq \ldots \end{align} $$

This sequence is easily seen to be a finite full flag, defined below.

The next theorem is Klyachko’s description of equivariant torsion-free sheaves. See also [Reference Perling30, Sec. 5.4] for this and a more general theorem.

Now, we can translate properties of ordinary sheaves into the language of these $\sigma $ -families. Recall that a sheaf F is reflexive if the canonical map $F \to F^{\vee \vee }$ of F into its double dual is an isomorphism. Every reflexive sheaf is torsion-free. On the other hand, using results from [Reference Huybrechts and Lehn10, Sec. 1.1], if F is torsion-free, then the map $F \to F^{\vee \vee }$ is injective and the quotient is zero-dimensional. Furthermore, it follows from [Reference Huybrechts and Lehn10, Sec. 1.1] that a coherent sheaf on a surface is reflexive if and only if it is locally free. Therefore, we state the following characterisation of equivariant reflexive sheaves on surfaces, which generalises to higher-dimensional smooth toric varieties.

In general, we only have the inclusion $\hat {F}_{\sigma }(n_1, n_2) \subseteq \hat {F}_\sigma (n_1, \infty ) \cap \hat {F}_{\sigma }(\infty , n_2)$ . Since we are dealing with finite $\sigma $ -families, we also know that equality fails only in a finite number of cases. This gives an easy description of $F^{\vee \vee }$ : We simply define $\hat {F}^{\vee \vee }(n_1, n_2)$ to be $\hat {F}(n_1, \infty ) \cap \hat {F}(\infty , n_2)$ . Then indeed $F^{\vee \vee }$ is reflexive by Proposition 3.7, and there is a natural inclusion $F \to F^{\vee \vee }$ with a cokernel that is finite-dimensional as a vector space (implying that it is supported in dimension zero (see [Reference Kool15, Prop. 2.8])).

As a corollary of the proposition, an equivariant vector bundle on X of rank n is completely determined by a finite complete flag for each ray of the fan $\Delta $ associated to X. An equivariant line bundle has an even easier description as a finite complete flag of $\mathbb {C}$ can be described by giving a number m: The flag is then given by $V_\lambda = 0$ if $\lambda < m$ and $V_\lambda = \mathbb {C}$ if $\lambda \geq m$ . Thus, to describe an equivariant line bundle, one needs to give an integer for each ray.

Example 3.8. The tangent bundle on $\mathbb {P}^2$ has a canonical equivariant structure. On the affine chart $\operatorname {\mathrm {Spec}} \mathbb {C}[\frac {X}{Z}, \frac {Y}{Z}]$ , the tangent sheaf is generated by $\frac {\partial }{\partial X/Z}$ and $\frac {\partial }{\partial Y/Z}$ . The action of $(s, t)$ on these generators is given by $s^{-1}$ and $t^{-1}$ respectively. Hence, the $\sigma $ -family on this cart can be pictured as follows. We have the $\mathbb {Z}^2$ -grid. The point with coordinates $(m, n)$ corresponds to the eigenspace of $s^mt^n$ . The vectors $\frac {\partial }{\partial X/Z}$ and $\frac {\partial }{\partial Y/Z}$ generate the weight spaces of weight $s^{-1}$ and $t^{-1}$ respectively. There are no nontrivial relations, so if $m, n$ are nonnegative we get the space $\mathbb {C}\frac {\partial }{\partial X/Z} \oplus \mathbb {C} \frac {\partial }{\partial Y/Z} \cong \mathbb {C}^2$ , and on the nodes $(-1, n)$ and $(m, -1)$ with m and n nonnegative we get $\mathbb {C}\frac {\partial }{\partial X/Z}$ and $\mathbb {C}\frac {\partial }{\partial Y/Z}$ respectively. One of the limiting flags is $\ldots \subseteq 0 \subseteq \mathbb {C}\frac {\partial }{\partial X/Z} \subseteq \mathbb {C}^2 \subseteq \ldots $ . One can similarly calculate the other flag and also do this for the other charts. The result is always the same: a flag $\ldots \subseteq 0 \subseteq \mathbb {C} \subseteq \mathbb {C}^2 \subseteq \ldots $ , where $\mathbb {C}$ has weight $-1$ .

We view the entire tangent bundle as a picture of a triangle with three ‘strips’; see Figure 1. The corners of the triangle represent the three charts of $\mathbb {P}^2$ . The center triangle corresponds to the part where the weight spaces have dimension 2. The strips correspond to the part where they have dimension 1. The case described above looked exactly like such a corner: There is an area, bounded on two sides, where the weight space has dimension 2. Furthermore, there were two strips where they have dimension 1. One can also see that they satisfy condition $(\star )$ , which is represented by the dotted lines.

Figure 1 The tangent bundle on $\mathbb {P}^2$ .

3.3 Chern classes of equivariant sheaves

Using the simple description of equivariant line bundles above it is easy to describe their classes in the Chow ring. This will allow us to compute the Chern characters of arbitrary equivariant torsion-free sheaves.

This lemma will allow us to compute all Chern characters of all equivariant sheaves encountered in this paper as it is generally very easy to find a resolution of an equivariant sheaf. The pictorial representation of the $\sigma $ -families is very helpful here. Below, we only work out a specific example, but it is easy to see how to construct more general resolutions in a similar manner. The abstract theory also ensures resolutions always exist; see [Reference Chriss and Ginzburg3]. Sometimes, it is also possible to construct such resolutions explicitly; see [Reference Perling30, Thm. 6.1].

3.4 Stability of equivariant bundles

Let X be a smooth toric variety with fixed polarisation H. We will give a criterion for an equivariant vector bundle E to be $\mu $ -stable on X. Note that, in general, one is interested in Gieseker semistable sheaves since this notion of stability gives the correct moduli space. We will only apply this criterion in situations where $\mu $ -stability and Gieseker stability coincide so that the criterion below is actually a criterion for Gieseker stability. We first introduce some notation.

Fix a toric variety X with polarisation H, and let E be an equivariant vector bundle of rank r over X. For each ray of $\Delta $ , the fan of X, we have a finite full flag (9). Recall that d is the number of rays of $\Delta $ , and let us denote for each $1 \leq i \leq d$ the flag associated to the i-th ray by $\hat {E}^i(\lambda )$ . For $1 \leq j \leq r-1$ , we let $\delta ^i_j$ be the number of spaces of dimension j in the flag $\hat {E}^i$ (which is always a finite number). If $W \subseteq \mathbb {C}^r$ is any subspace, denote by $w^i_j$ the dimension of the intersection of W with some $\hat {E}^i(\lambda )$ of dimension j. This does not depend on the choice of $\lambda $ , and if no such $\lambda $ exists, the number $w^i_j$ will not be important and can have arbitrary value. Finally, to each ray i corresponds a divisor $D_i$ . We define $\deg D_i$ as $H.D_i$ .

The next criterion is established in the course of the proof of Theorem 3.20 (for $\mu $ -semistability) and in Proposition 4.13 (for $\mu $ -stability) in [Reference Kool15].

Proposition 3.11. An equivariant vector bundle E of rank r is $\mu $ -stable if and only if for each nontrivial subspace $0 \subsetneq W \subsetneq \mathbb {C}^r$ the following inequality holds:

$$ \begin{align*} \frac{1}{\dim W} \sum_{i = 1}^d \sum_{j = 1}^{r - 1} \delta^i_j \cdot \deg D_i \cdot w^i_j < \frac{1}{r} \sum_{i = 1}^d \sum_{j = 1}^{r - 1} \delta^i_j\cdot \deg D_i\cdot j. \end{align*} $$

For $\mu $ -semistability, replace $<$ by $\leq $ in the above inequality.

3.5 Equivariant K-theory

In this paragraph, we describe a localisation formula for K-theory. Let X be a smooth projective variety on which a torus T acts. Let $X^T$ be the fixed point locus with inclusion $\iota : X^T \to X$ . Then $X^T$ is smooth as well. Let N be the normal bundle of $X^T$ in X. Denote by $K_T^0(X)$ and $K_T^0(X^T)$ their equivariant K-theories. Then there is a localisation theorem, which was first proven in [Reference Thomason33]. The following formulation can be found in [Reference Okounkov23, Sec. 2.3].

Theorem 3.14. There exists finitely many characters $\mu _i$ of T such that the pushforward map $\iota _* : K_T^0(X^T) \to K_T^0(X)$ becomes an isomorphism after localising at $1 - \mu _i$ for all i. Furthermore, in this case the class $\bigwedge ^{\bullet } N^{\vee } \in K^0_T(X^T)$ becomes invertible and we have that

$$ \begin{align*} F = \iota_*\bigg(\frac{\iota^{*}F}{\bigwedge^{\bullet} N^{\vee}}\bigg) \end{align*} $$

for all F in the localised equivariant K-theory of X.

If X is a toric variety, then $X^T$ is a disjoint union of reduced points, one for each two-dimensional cone in the fan of X. The K-theory of a point is the ring of representations. When X is two-dimensional and $T = \mathbb {G}_m^ 2$ , this ring is $\mathbb {Z}[s, t]$ . Hence, $K^0_T(X^T) = \prod _{p \in X^T}\mathbb {Z}[s,t]$ .

In this case, Theorem 3.14 takes the following form. Since $X^T$ is zero-dimensional, $N = T_X|_{X^T}$ . For each point $p \in X^T$ , we have that $\bigwedge ^\bullet N^\vee |_{\{p\}} = \bigwedge ^{\bullet } \Omega _X|_{\{p\}}$ . If we write $\Omega _X|_{\{p\}} = \chi _{p, 1} + \chi _{p,2}$ in the representation ring, then $\bigwedge ^\bullet \Omega _X|_{\{p\}} = (1 - \chi _{p,1})(1 - \chi _{p,2})$ .

We will be interested in computing the Euler characteristic of a sheaf using localisation. For this, we push the equality of Theorem 3.14 to a point, which gives us:

(10) $$ \begin{align} \chi(X, F) = \sum_{p \in X^T} \frac{F|_{\{p\}}}{(1 - \chi_{p,1})(1 - \chi_{p,2})}. \end{align} $$

3.6 Equivariant cohomology

Here, we recall Atiyah–Bott localisation, which we use to evaluate integrals on the moduli space M. The theory is similar to the equivariant K-theory described above. Let X be a smooth projective variety on which a torus T acts. Again, this implies that the fixed point locus $X^T$ is smooth. Let $\iota : X^ T \to X$ be the inclusion and N be the normal bundle. We have equivariant cohomology groups $H^\bullet _T(X, \mathbb {Q})$ and $H^\bullet _T(X^T, \mathbb {Q})$ . Then we have the following theorem:

Theorem 3.15 [Reference Edidin and Graham4, Thm. 2]

For any equivariant cohomology class $\alpha \in H^\bullet _T(X, \mathbb {Q})$ , we have the equality

$$ \begin{align*} \int_X \alpha = \int_{X^T} \frac{\iota^* \alpha}{e(N)}. \end{align*} $$

Here, $e(N)$ denotes the Euler class of the normal sheaf N. This equality holds after inverting all characters of positive degree.

In the cases where we apply this theorem, $X^T$ is isolated. Then we have that $N = T_X|_{X^T}$ . Then the integral to the right becomes a finite sum, which is easier to evaluate.

This theorem can also be used to compute nonequivariant integrals. For this, we use that there is a forgetful map $H^\bullet _T(X, \mathbb {Q}) \to H^\bullet (X, \mathbb {Q})$ . To compute the integral of a cohomology class $\alpha $ in $H^\bullet (X, \mathbb {Q})$ , if we have a lift $\alpha ' \in H^\bullet _T(X, \mathbb {Q})$ , then we can use Theorem 3.15 to evaluate the integral $\int _X \alpha ' \in H^\bullet _T(\{*\}, \mathbb {Q})$ . One can then apply the forgetful map to obtain the integral of $\alpha $ .

4.1 The computation

We will explain how to evaluate the equivariant integral $\int _M P$ , where P is any polynomial in the $\operatorname {\mathrm {ch}}_i(\gamma )$ , for $\gamma $ an equivariant cohomology class on X. One can set $P = \mathcal {L}_kD$ to verify the conjecture. In particular, we will see that the only input we need is $T_{X, p}$ for $p \in X^T$ as two-dimensional representation of T and $\mathcal {E}_{p, q}$ , the equivariant K-theory class of the equivariantFootnote ⁴ sheaf $\mathcal {E}_q$ corresponding to $q \in M^T$ at $p \in X^T$ . This is a finite amount of discrete data. The author has used the freely available computer algebra system SageMath [32] to run the computation. The code is available on his website.

First, we compute $\operatorname {\mathrm {ch}}_i(\gamma )$ for any i and $\gamma $ as elements of $H^\bullet _T(M^T, \mathbb {Q})$ . Since $X^T \times M^T$ is zero-dimensional, the map $\operatorname {\mathrm {ch}} : K_T(X^T \times M^T) \to H_T^\bullet (X^T \times M^T, \mathbb {Q})$ is an isomorphism, so the component of $\operatorname {\mathrm {ch}}(\mathcal {E})$ at the point $(p, q) \in X^T \times M^T$ is simply $\mathcal {E}_{p, q}$ . We can then compute the classes (3) by using a formal power series as mentioned in the introduction. Since M is smooth of the expected dimension $d = 2rc_2 - (r - 1)c_1^2 - (r^2-1)\chi (X, \mathcal {O}_X)$ , we can safely ignore the terms in the power series of degree higher than $2d$ , so this becomes a finite sum. Now, given $\gamma $ with component $\gamma _p$ at $p \in X^T$ , localisation allows us to compute:

$$ \begin{align*} \operatorname{\mathrm{ch}}_i(\gamma)_q = \sum_{p \in X^T} \frac{\gamma_p \cdot \operatorname{\mathrm{ch}}_i\bigg(-\mathcal{E}_{p,q} \otimes \det(\mathcal{E}_{p, q})^{-1/r}\bigg)}{\text{e}(T_{X, p})}. \end{align*} $$

This allows us to compute any polynomial in the $\operatorname {\mathrm {ch}}_i(\gamma )$ , simply by addition and multiplication. If such a polynomial has component $P_q$ at $q \in M^T$ , localisation tells us that we want to consider

$$ \begin{align*} \int_M P = \sum_{q \in {M^ T}} \frac{ P_q }{ \text{e}(T_{M, q}) }. \end{align*} $$

This reduces the problem to computing $T_{M, q}$ as representation of T. It is well known that