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 $T_{M, q} \cong \operatorname {\mathrm {Ext}}^1(E_q, E_q)_0$ , the trace-free Ext-group, and this isomorphism respects the T-structure. The codomain of the trace map is $H^1(X, \mathcal {O}_X)$ , which is zero for a toric surface. Hence, we find $\operatorname {\mathrm {Ext}}^1(E_q, E_q)_0 = \operatorname {\mathrm {Ext}}^1(E_q,E_q)$ . Since $\operatorname {\mathrm {Ext}}^0(E_q, E_q) = 1$ (as representation) and $\operatorname {\mathrm {Ext}}^2(E_q, E_q) = 0$ by Lemma 4.1, $\operatorname {\mathrm {Ext}}^1(E_q, E_q) = 1 - \chi (E_q, E_q)$ as elements of the representation ring. The Euler class is a K-theoretic invariant, which can be computed by the K-theoretic localisation formula (10). This gives us

$$ \begin{align*} \chi(E_q, E_q) = \sum_{p \in X^T} \frac{\mathcal{E}_{p, q} \otimes \mathcal{E}_{p, q}^\vee}{(1 - \chi_{p,1})(1 - \chi_{p,2})}, \end{align*} $$

where $\chi _p^1$ and $\chi _p^2$ are as in equation (10). This completes the calculation. It is evident from the computation that this can be done by a computer, given the data specified before.

Example 4.2. Recall that on $X = \mathbb {F}_0 = \mathbb {P}^1\times \mathbb {P}^1$ , we have that $H^2(X, \mathbb {Q})$ is generated by F, Z, where F is a fibre of the projection $\pi _2 : \mathbb {P}^1 \times \mathbb {P}^1 \to \mathbb {P}^1$ and Z is the image of a section of $\pi _2$ . The fan of X has four cones of dimension two, which are each given by a quadrant of $\mathbb {Z}^2$ . We number the associated fixed points by letting $X_1$ be the fixed point corresponding to the upper-right quadrant and then proceeding clockwise to define $X_2$ , $X_3$ and $X_4$ . Then the tangent spaces at these points have representations $s^{-1} + t^{-1}$ , $s^{-1} + t$ , $s + t$ and $s + t^{-1}$ respectively, where $\mathbb {Z}[s, t]$ is the representation ring of the two-dimensional torus.

Consider the case where $r = 2$ , $\Delta = F+Z$ , $c_2 = 2$ and with polarisation $H = 2F + 5Z$ . Then the fixed point locus consists of four equivariant sheaves. In Table 1, we put what the representation of such a sheaf F is when restricted to a fixed point $X_i$ .

Table 1 Equivariant data of some sheaves on $\mathbb {F}_0$ .

Of course, one has to make a choice of equivariant structure. We explain a way of doing so when explaining how to find all vector bundles.

As the reader can see, all the elements in the table are honest representations, no minuses occur. This is a sign that all of the sheaves are vector bundles, which is indeed the case. Using K-theoretic localisation (10), we can compute that the tangent spaces to M at these sheaves are respectively $t^{-1} + s^{-1} + s^{-1}t^{-1}$ , $st + s + t$ , $s + st^{-1} + t^{-1}$ and $t + ts^{-1} + s^{-1}$ . Their Euler classes are therefore $-s^2t - st^2$ , $s^2t + st^2$ , $-s^2t + st^2$ and $s^2t - st^2$ .

4.2 Verifying the conjecture

Next, we explain how to verify the conjecture given the algorithm above and the data $\mathcal {E}_{p, q}$ and $T_{X, p}$ . Fix a basis $\gamma _k$ of $H^\bullet (X, \mathbb {Q})$ , where each $\gamma _k$ is of pure degree. It suffices to verify the conjecture for $D = \prod _j \operatorname {\mathrm {ch}}_{i_j}(\gamma _{k_j})$ a monomial, where each $\operatorname {\mathrm {ch}}_{i_j}(\gamma _{k_j})$ has positive degree. Indeed, if one of the $\operatorname {\mathrm {ch}}_{i_j}(\gamma _{k_j})$ has nonnegative degree we can pull them out of $\mathcal {L}_k$ by Lemma 2.2. Hence, we can restrict to this case.

But there are essentially a finite amount of such D since the conjecture is also automatic if $\deg D> \dim M$ by Corollary 2.3. But one can generate a list of all D with a fixed degree. Hence, we should, for each k, generate a list of all D with degree $\dim M - k$ and check the conjecture for $\mathcal {L}_kD$ .

To perform this check for a given monomial D, choose equivariant lifts $\beta _k$ for the $\gamma _k$ . Then we perform the computation described above. The result lives in $H^\bullet _T(\{*\}, \mathbb {Q}) = \mathbb {Q}[s, t]$ , but we can plug in $s = t = 0$ to obtain a rational number. This is the desired integral $\int _M \mathcal {L}_kD$ .

Example 4.5. We continue with Example 4.2. We first choose equivariant lifts of the cohomology classes F and Z. It suffices to give these lifts at the fixed points $X_i$ . For a single point, we have $H^\bullet _T(\{*\}, \mathbb {Q}) \cong K^0_T(\{*\}) \otimes \mathbb {Q}= \mathbb {Q}[s, t]$ . It turns out that there is an equivariant class lifting F which restricts to $s^{-1}$ at $X_1$ and $X_2$ and zero otherwise, and there is one lifting Z which restricts to $t^{-1}$ at $X_1$ and $X_4$ and zero otherwise. Then the point class $\mathbf {p}$ is simply the product of these classes.

Now, we can compute all classes $\operatorname {\mathrm {ch}}_i(\gamma )$ , but to verify the conjecture, one also needs the operator $\mathcal {L}_k$ . For this, we also need a Künneth decomposition of the diagonal $X \to X \times X$ . This can be computed to be $\mathbf {p} \otimes 1 + F \otimes Z + Z \otimes F + 1 \otimes \mathbf {p}$ , for example, by the method of [Reference Eisenbud and Harris6, Sec. 2.1.6].

For example, to verify that the conjecture holds for $\mathcal {L}_2\operatorname {\mathrm {ch}}_2(Z)$ , one needs to integrate

$$ \begin{align*}R_2\operatorname{\mathrm{ch}}_2(Z) = 6\operatorname{\mathrm{ch}}_4(Z), \qquad T_2\operatorname{\mathrm{ch}}_2(Z)\quad \text{and} \quad S_2\operatorname{\mathrm{ch}}_2(Z) = 3(\operatorname{\mathrm{ch}}_2(\mathbf{p})\operatorname{\mathrm{ch}}_2(Z) + \operatorname{\mathrm{ch}}_3(\mathbf{p})\operatorname{\mathrm{ch}}_1(Z)) \end{align*} $$

We refrain from writing down all intermediate steps and simply give the result: $-\frac 18$ , $-\frac {1}{4}$ and $\frac 38$ . These sum up to zero, as required.

4.3 Finding all equivariant sheaves

Above, we assumed we already had access to all equivariant stable sheaves with certain fixed invariants. Now, we explain how to find these. We assume that we have a fixed polarisation H. Later, we will explain how to find all possible H so that one can apply this procedure to each one. The procedure can be divided into two steps: First, find all equivariant vector bundles, which we then use to find all equivariant torsion-free sheaves. We first explain how to reduce to vector bundles.

4.3.1 Reducing to vector bundles

Note that, for any $\mu $ -stable sheaf E with fixed invariants r, $c_1$ and $c_2$ , the canonical map $E \to E^{\vee \vee }$ is an embedding of E into a $\mu $ -stable vector bundle of rank r, $c_1(E^{\vee \vee }) = c_1$ and $c_2(E^{\vee \vee }) \leq c_2$ . But we also have the lower bound $c_2(E^{\vee \vee }) \geq \frac {(r - 1)c_1^2}{2r}$ given by the Bogomolov inequality. Thus, if we are able to find a list of all stable equivariant vector bundles F satisfying $\operatorname {\mathrm {rk}} F = r$ , $c_1(F) = c_1$ and $\frac {(r - 1)c_1^2}{2r} \leq c_2(F) \leq c_2$ , then $E^{\vee \vee }$ must be in this list. Furthermore, if we have such an equivariant vector bundle F and any equivariant torsion-free sheaf E with $E^{\vee \vee } = F$ , then E is $\mu $ -stable. Thus, it suffices to, given F, find all equivariant torsion-free E with the right invariants such that $E^{\vee \vee } = F$ .

Consider the following operation on F. We consider a maximal cone $\sigma $ and the $\sigma $ -family $\hat {F}$ of F on $U_{\sigma }$ . We consider a point $(m, n)$ such that $\hat {F}(m-1, n) + \hat {F}(m, n-1)$ is a proper subspace of $\hat {F}(m, n)$ . We then replace $\hat {F}(m, n)$ by a subspace of codimension one, which contains $\hat {F}(m-1, n)+\hat {F}(m, n-1)$ . This gives a subequivariant sheaf $F'$ of F which has its $c_2$ increased by one. Since $(F')^{\vee \vee } = F^{\vee \vee }$ , $F'$ is still $\mu $ -stable and hence Gieseker stable. We have constructed $F'$ in such a way that there is a short exact sequence

(11) $$ \begin{align} 0 \to F' \to F \to \chi \cdot \mathcal{O}_{p} \to 0. \end{align} $$

Here, p is the fixed points of X corresponding to the maximal cone $\sigma $ and $\chi $ is the character of $F(m, n)$ .

All equivariant sheaves E with $E^{\vee \vee } = F$ are obtained by applying this operation sufficiently many times so that the resulting sheaf has the desired $c_2$ . If the difference in $c_2$ ’s is only one, this can easily be seen by considering the quotient. Otherwise, one can use induction.

Example 4.6. On $\mathbb {P}^2$ there are 1, 3 and 3 equivariant vector bundles of rank 2 with $c_1 = 1$ and $c_2$ equal to respectively 1, 2 and 3. Each stable equivariant bundle with $c_2 = 2$ degenerates in six ways to a stable equivariant sheaf with $c_2 = 3$ . Thus, the bundles with $c_2= 2$ give a contribution of $3 \cdot 6 = 18$ stable equivariant sheaves. The single equivariant bundle with $c_1 = 1$ degenerates to a total of 27 stable equivariant sheaves with $c_2 = 3$ . Hence, the moduli space has $3 + 18 + 27 = 48$ isolated fixed points, the highest of the examples in this paper. See Appendix A for a complete list.

We make this more concrete in a specific example. There is a stable equivariant vector bundle E with $c_2 = 2$ whose K-theory classes at the fixed points of $\mathbb {P}^2$ are $(st^{-1} + ts^{-1}, st^{-1} + t, ts^{-1} + s)$ . To perform the operation described above, we can consider the first chart and the weight space of $st^{-1}$ . This space is one-dimensional, so there is no choice in picking a codimension-one subspace. We then obtain an equivariant stable torsion-free sheaf $E'$ which is not locally free.

In this paper, we are mainly interested in the K-theory class of $E'$ . For this, we can use the exact sequence (11). It is not difficult to see that the K-theory class of $\chi \cdot \mathcal {O}_p$ is $\chi \cdot (1 - s - t + st)$ . So we can obtain the K-theory class of $E'$ by replacing $st^{-1}$ by $st^{-1}(s + t - st)$ in the K-theory class of E. This computation also works if we had chosen the weight space $ts^{-1}$ . It also works if we had chosen one of the other two charts, except that the K-theory class of $\chi \cdot \mathcal {O}_p$ is then $\chi \cdot (1 - s^{-1} - ts^{-1} + ts^{-2})$ or $\chi \cdot (1 - t^{-1} - st^{-1} + st^{-2})$ respectively.

Example 4.7. We show how to represent the equivariant torsion-free sheaves pictorially on a Hirzebruch surface. Recall from Example 3.8 that one can represent a rank 2 vector bundle on $\mathbb {P}^2$ by a triangle with strips attached. For a Hirzebruch surface, the picture is the same, except that we now have a square instead of a triangle (essentially because on a Hirzebruch surface we have four toric charts). Each of the corners represents one of the charts and each side represents a ray of the cone (i.e., the gluing condition $(\star )$ ). The first picture in Figure 2 represents an equivariant vector bundle on $\mathbb {F}_a$ . Consider the chart in the upper-right corner. In this chart, there are two places where we can apply the operation described above, namely in the upper-right corners of the two strips. Suppose we choose the right strip. Then we lower the dimension of the weight space in the upper-right corner from one to zero. Hence, we obtain the second picture in Figure 2. The small square we removed represents the single weight space we removed.

Figure 2 Three equivariant sheaves on a Hirzebruch surface.

4.4 Finding all equivariant vector bundles

We describe how to find all equivariant stable vector bundles. We focus on the rank 2 case on a Hirzebruch surface $\mathbb {F}_a$ , but the other cases are similar. The fan of a Hirzebruch surface has four two-dimensional cones and four rays and can be found in [Reference Fulton7, Sec. 1.1]. We number the rays in clockwise order, starting with the rightmost ray. In this case, an equivariant vector bundle E is described by four finite full flags of $\mathbb {C}^2$ , one for each ray. Thus, for each ray $\rho _i$ we have a sequence which looks like the following:

$$ \begin{align*} \ldots \subseteq 0 \subseteq \ldots \subseteq V^i \subseteq \ldots \subseteq V^i \subseteq \ldots \subseteq \mathbb{C}^2 \subseteq \ldots \end{align*} $$

Such a flag depends on three choices: the one-dimensional subspace $V^i$ , a number $\delta ^i$ which is the number of occurences of $V^i$ and a number $A^i$ which is the weight where $\mathbb {C}^2$ first occurs. However, different choices may lead to the same vector bundle since we have the freedom of tensoring with a character from $T = \mathbb {G}^2_m$ and we can apply a linear automorphism of $\mathbb {C}^2$ to obtain different $V^i$ . The action of a character of T only changes the $A^i$ . To be precise, the action of $s^nt^m$ changes $(A^1, A^2, A^3, A^4)$ to $(A^1+n, A^2 - m, A^3 - n, A^4 + m)$ . So we can eliminate the ambiguity of the choice of character by fixing (for example) $A^2 = A^3 = 0$ . The ambiguity in the choice of $V^i$ is more subtle. For the moment, let us assume that all the $V^i$ are distinct. The computation does not depend on the specific choice of $V^i$ . Later, we will deal with the other cases. Pictorially, this means we are in the situation corresponding to the first picture in Figure 2.

We will determine all possible numerical invariants. These are the solutions to a certain set of equalities and inequalities, determined by fixing $c_1$ and $c_2$ , and the inequalities coming from stability (Proposition 3.11). By considering a resolution, we can find explicit formulas for the first and second Chern character of E. The formulas also depends on a (recall we work on $\mathbb {F}_a$ ).

$$ \begin{align*} \operatorname{\mathrm{ch}}_1(E) = (\delta^1 + \delta^3 + a\delta^2 - 2A^1)F + (\delta^2 + \delta^4 - 2A^4)Z. \end{align*} $$

Hence, if we have fixed $\Delta = fF + zZ$ , then we get the equations $f = \delta ^1 + \delta ^3 + a\delta ^2 - 2A^1$ and $z = \delta ^2 + \delta ^4 - 2A^4$ . The formula for the second Chern character is

(12) $$ \begin{align} \operatorname{\mathrm{ch}}_2(E) = \frac14 \big( a((\delta^2)^2 - (\delta^4)^2 - z^2) - 2(\delta^1\delta^2 + \delta^2 + \delta^3 + \delta^3 \delta^4 + \delta^4 \delta^1) + 2fz \big). \end{align} $$

This formula requires the $V_i$ to be distinct.

Next, we need to consider the stability inequalities from Proposition 3.11. To compute the degrees, we need the intersection numbers $\deg _1 = \deg _3 = H.F$ , $\deg _2 = H.(aF + Z)$ and $\deg _4 = H.Z$ . The proposition tells us that we need to consider all possible one-dimensional subspaces W, but in fact it suffices to consider $W = V^i$ for some $V^i$ . Indeed, otherwise the inequality is trivial as $w^i_j$ is always zero in that case. Then the stability inequalities become

(13) $$ \begin{align} 2\delta^i \deg_i < \sum_{j = 1}^4 \delta^j \deg_j \end{align} $$

for each i. Here, we also use that the $V^i$ are distinct. Finally, we need the trivial inequalities $\delta _i \geq 0$ . We then find all solutions for this system. This is elementary, but still difficult. We made use of the automated theorem prover Z3 [Reference de Moura and Bjørner22] to find all solutions.

Now, we deal with the problem of choice of the vector spaces $V^i$ . It turns out that, in the numbers we obtained above, at least one $\delta ^i$ was always zero. Thus, we only had to choose three distinct vector spaces. But then there is no choice at all: If we choose different subspaces $W^i$ , then there is a linear automorphism of $\mathbb {C}^2$ mapping $V^i$ to $W^i$ . So the resulting equivariant stable vector bundle does not depend on the choice of $V^i$ . This also implies that the fixed point locus $M^T$ is isolated since it only depends on discrete numerical data. If all $\delta ^i$ are positive, we would have had a higher-dimensional component of $M^T$ . However, in this paper, it turns out that this behaviour does not occur and we only have to deal with cases where $\dim M^T = 0$ .

4.5 Degenerate cases

In the previous computation, we assumed that the $V^i$ were all distinct. Here, we explain how to find the vector bundles if that were not true. The stability inequality Proposition 3.11 implies the following: There must always exist at least three i for which $\delta ^i> 0$ and for which the $V^i$ are distinct. Then there are the following cases to consider: Two adjacent $V^i$ are equal (e.g. $V^1 = V^4$ ) or two opposing $V^i$ are equal (e.g., $V^1 = V^3$ ). We explain the changes.

The formula (12) for the second Chern character changes if two adjacent $V^i$ are equal. Explicitly, if $V^1 = V^4$ , one should add a term $\delta ^1\delta ^4$ to the formula and similarly if other adjacent $V_i$ are equal. This is similar for the other cases. The other thing that changes is the stability conditions, both in the case of adjacent and opposing coincidence of the $V^i$ . Indeed, suppose $V^1 = V^3$ , then taking $W = V^1 = V^3$ in Proposition 3.11 gives the more restrictive inequality

(14) $$ \begin{align} 2(\deg_1\delta^1 + \deg_3 \delta^3) < \sum_{j = 1}^4 \deg_j\delta^j. \end{align} $$

One needs to add this inequality to the system we already had in equation (13). We can solve this system in a similar way to the other case. Because there are only three different vector spaces $V^i$ from the start, we can argue in the same way as before to ensure that $M^T$ is isolated.

Pictorially, the condition that $V^1 = V^4$ corresponds to the third picture in Figure 2 where we have filled in the corner. Indeed, if we consider this chart, then recall that for a vector bundle E we should have $\hat {E}(m, n) = \hat {E}(m, \infty ) \cap \hat {E}(\infty , n)$ . If $(m, n)$ lies in this corner, this intersection is $V^1 \cap V^4$ . In a generic situation, $V^1 \neq V^4$ so that the intersection is zero. But in this special case, the intersection equals $V^1 = V^4$ , so it is one-dimensional.

The picture also explains why the second Chern character changes. The little square we added by setting $V^1 = V^4$ has size $\delta ^1 \times \delta ^4$ . It turns out that the Chern character only depends on the dimension of the weight spaces; see [Reference Kool14, Prop. 3.6]. Changing a single weight space by one dimension changes the second Chern character by one, and the other Chern characters stay the same. Therefore, a square of dimension $\delta ^1 \times \delta ^4$ represents a change of $\delta ^1\delta ^4$ in the second Chern character. Note that, in the situation $V^1 = V^3$ , we do not add such a small square, and consequently, the second Chern character does not change.

4.6 Finding bundles on $\mathbb {P}^2$

Finding stable equivariant vector bundles for $\mathbb {P}^2$ is similar to what we have just described. Recall that the fan for $\mathbb {P}^2$ has three rays. Thus, the rank 2 case becomes even easier since there are only three vector spaces $V^i$ from the start, which makes it easier to argue that $M^T$ is isolated. However, the rank 3 and 4 cases are much more involved. We briefly discuss the rank 3 case. Now, our flags look like

$$ \begin{align*} \ldots \subseteq 0 \subseteq V^i \subseteq \ldots \subseteq V^i \subseteq W^i \subseteq \ldots \subseteq W^i \subseteq \mathbb{C}^3 \subseteq \ldots \end{align*} $$

where $V^i$ is one-dimensional and $W^i$ is two-dimensional. Note that we require three numbers to describe such a flag, with $A^i$ as before and $\delta ^i = \delta ^i_1$ and $\epsilon _i = \delta ^i_2$ for the number of occurrences of $V^i$ and $W^i$ , respectively. Having two spaces in our flag gives a great additional complexity in the ambiguity of choices of these subspaces. Also, classifying the possible coincidences is more complicated. This was done by Klyachko in a preprint [Reference Klyachko13], but see also [Reference Kool14, Sec. 4.2]. Recall that one- and two-dimensional subspaces of $\mathbb {C}^3$ are the points and lines in $\mathbb {P}^ 2$ . Thus, we may picture the $V^i$ and $W^i$ as three lines with three points on them (indicating that the $V^i$ are subspaces of the $W^i$ ). Then stability ensures that the possible configurations are the ones pictured in Figure 3.

Figure 3 Possible coincidences.

The resulting inequalities are also much harder than in the Hirzebruch case. We applied a mix of calculations by hand and the solver Z3 to find all solutions. In the end, we can argue as before that, since a sufficient number of $\delta ^i$ and $\epsilon ^i$ vanish, the isomorphism class of the obtained vector bundle $E^i$ does not depend on the choice of subspaces $V^i$ and $W^i$ as any two choices can be related by a suitable automorphism of $\mathbb {C}^3$ . This implies that $M^T$ is isolated, as before.

The classification of bundles of rank 4 on $\mathbb {P}^2$ is incomplete as we found it already too difficult to classify the possible coincidences of the subspaces in the flags in this case. We found 13 bundles by looping over the $\delta ^i$ in the nondegenerate case where there are no special coincidences. We are nevertheless confident that there are no more bundles because the Atiyah–Bott calculation gave numbers as a result. Usually, when one forgets a bundle, the answer is a not a number but a more complicated expression in the equivariant parameters.

4.7 Dependence on the polarisation

Both $\mu $ -stability and Gieseker stability depend on the choice of a polarisation. We are interested in verifying the conjecture for all possible choices of polarisation. For $\mathbb {P}^2$ , this is not an issue since the only polarisations are $\mathcal {O}(n)$ for $n> 0$ and the stability condition does not change if you scale a polarisation by a positive number.

But for Hirzebruch surfaces this is an issue. Note that $\operatorname {\mathrm {Pic}}(\mathbb {F}_a) = \operatorname {\mathrm {Num}}(\mathbb {F}_a) = \mathbb {Z}F \oplus \mathbb {Z}Z$ with intersections $F.F = 0$ , $F.Z = 1$ and $Z.Z = -a$ . The ample cone is the set of line bundles L such that $L.F> 0$ and $L.Z> 0$ . $\operatorname {\mathrm {Num}}(\mathbb {F}_a)$ contains a finite set of hyperplanes, called walls, which divide the ample cone into connected components, called chambers. This material can be found in [Reference Huybrechts and Lehn10, Sec. 4.C] and was developed by Z. Qin [Reference Qin31]. If F is a rank 2 sheaf F with $c_1(F)$ not divisible by 2, then F is $\mu $ -stable with respect to a polarisation H iff it is $\mu $ -stable with respect to any polarisation in the chamber of H. Hence, the moduli space $M = M^H_X(r, \Delta , c_2)$ does not change if we pick another polarisation in the same chamber. We do not need to consider polarisations H that lie on walls: If the moduli space M changes if we pick another polarisation $H'$ that lies close to H and is in a chamber, then this change happens because of the existence of strictly $\mu $ -semistable sheaves with respect to H. But we assumed no such sheaves existed.

Now, we explain how to find all walls. For a general surface X and fixed r, $\Delta $ and $c_2$ , consider all the $\xi \in \operatorname {\mathrm {Num}}(X)$ such that

$$ \begin{align*} \frac{((r^2 - 1)\Delta^2 - 2rc_2)r^2}{4} \leq \xi^2 < 0.\end{align*} $$

Then the walls are given by the hyperplanes $\{\xi \}^\perp $ . Furthermore, if X is a Hirzebruch surface, there are only a finite number of such $\xi $ . To verify Conjecture 1.4, it suffices to choose a polarisation from each of the chambers and apply the algorithm described in the beginning of this section.

Example 4.8. In case $X = \mathbb {F}_0$ , $r = 2$ , $\Delta = F$ and $c_2 = 2$ , the choice of polarisation is relevant. There there are eight chambers, which we can represent by the following polarisations:

$$ \begin{align*} F + 5Z,\; 2F + 7Z,\; 2F + 5Z,\; 3F + 5Z,\; 6F + 5Z,\; 11F + 5Z,\; 16F + 5Z,\; 21F + 5Z. \end{align*} $$

For the first polarisation, the moduli space is empty. The next two are in different chambers, yet we found that they give rise to the same fixed locus. Theoretically, it is possible that the moduli spaces are different, but this seems unlikely. In this case, there are six vector bundles in the fixed locus. The final five polarisations also share the same fixed locus. Two of the vector bundles of the previous case are still stable here. But now there are 24 other sheaves: 20 of which are only torsion-free, not vector bundles, and four new vector bundles which are degenerate in the sense of Section 4.5. Thus, even the topological Euler characteristic depends on the choice of polarisation (as is well known). All the sheaves are listed in Appendix A.

These two examples are the only cases treated in this paper in which the polarisation is relevant.

4.8 A deformation argument

Using the algorithm above, we have verified Conjecture 1.4 explicitly for the Hirzebruch surfaces $\mathbb {F}_a$ with $a = 0, 1, 2$ , $\Delta = F$ , Z and $H+Z$ and $c_2$ the minimal choice such that the Bogomolov inequality is satisfied. We will now use a deformation argument to deduce Conjecture 1.4 for any a with $\Delta $ and H such that $\Delta .H$ is odd and $c_2$ again the minimal choice. In fact, the case $a = 2$ is redundant as it also follows from the deformation argument.

Proof. We work on $\mathbb {P}^1$ . Consider the subsheaf $L_1$ of $\mathcal {O}$ of functions that have a zero at 0 of order at least n. Similarly, we consider the subsheaf $L_2$ of functions that have a zero at $\infty $ of order at least n. The sum of the inclusions $L_1 \oplus L_2 \to \mathcal {O}$ is a surjection whose kernel consists of the functions vanishing at both zero and $\infty $ of order at least n. Of course, $L_1 \cong L_2 \cong \mathcal {O}(-n)$ and the kernel is $\mathcal {O}(-2n)$ . Hence, this gives a short exact sequence

$$ \begin{align*} 0 \to \mathcal{O}(-2n) \to \mathcal{O}(-n) \oplus \mathcal{O}(-n) \to \mathcal{O} \to 0. \end{align*} $$

Let $\mathbb {A}^1$ be the line in $\operatorname {\mathrm {Ext}}^1(\mathcal {O}, \mathcal {O}(-2n))$ through zero and the above extension. Then there is a $\mathbb {A}^1$ -flat family $\mathcal {E}$ on $\mathbb {P}^1 \times \mathbb {A}^1$ , which is $\mathcal {O}(-n) \oplus \mathcal {O}(-n)$ over every nonzero fibre and $\mathcal {O}(-2n) \oplus \mathcal {O}$ over zero [Reference Lange16, Rem. 3.5]. Let $\mathcal {F}$ be the projectivisation of $\mathcal {E}$ . Then $\mathcal {F}$ satisfies the conditions of the lemma. For the odd case, a similar argument works.

Proof. The idea is to use Proposition 2.14. We have constructed a smooth family $\mathcal {F} \to \mathbb {A}^1$ in Lemma 4.10. By Lemma 4.11, we can extend the polarisation H and the line bundle $\Delta $ to families $\mathcal {H}$ and $\mathcal {L}$ on $\mathcal {F}$ . We can also extend $c_2$ by taking a suitable multiple of (the extended version of) $F.Z$ . The intersection numbers $\mathcal {H}_t. \mathcal {L}_t$ are the same for every fibre. This is also true for other intersection numbers. Hence, we deduce:

1. 1. $\mathcal {H}_t.\mathcal {L}_t$ is odd for each closed point $t \in \mathbb {A}^1$ .

2. 2. $\mathcal {H}_t$ is ample for each closed point $t \in \mathbb {A}^1$ , hence is relatively ample for $\mathcal {F} \to \mathbb {A}^1$ [Reference Lazarsfeld17, Sec. 1.7].

3. 3. $c_2$ is the minimal $c_2$ such that the Bogomolov inequality holds with $r = 2$ and $c_1 = \mathcal {L}_t$ .

We wish to verify the Virasoro constraints for the central fibre, so by Proposition 2.14 we can also do so for another fibre, which is always $\mathbb {F}_0$ or $\mathbb {F}_1$ . We treat the first case as the second is similar. For concreteness, we pick the fibre over $1 \in \mathbb {A}^1$ and verify Conjecture 1.4 there. Since $\mathcal {H}_1.\mathcal {L}_1$ is odd, $\mathcal {L}_1$ is not a multiple of 2. Hence, there is a line bundle $L'$ on $\mathbb {F}_0$ such that $\mathcal {L}_1 + 2 L'$ is either F, Z or $F+Z$ . Now, we use that, when $\mu $ -stability and Giesker stability coincide, there is an isomorphism

$$ \begin{align*}M^{\mathcal{H}_1}_{\mathbb{F}_0}(2, \mathcal{L}_1, c_2) \to M^{\mathcal{H}_1}_{\mathbb{F}_0}(2, \mathcal{L}_1 + 2L', c_2 + L'.(\mathcal{L}_1 + L')),\end{align*} $$

which is given by $[E] \mapsto [E \otimes L']$ on closed points. On this second space, $c_2$ is still the minimal number such that the Bogomolov inequality holds. Hence, we already know the Virasoro constraints for this case by our explicit computation.