2.1 The stack of coherent sheaves on a surface $S$

Let $S$ be a smooth connected quasi-projective surface. Unless mentioned otherwise, we will make the following assumption:

\begin{align*} \boxed{{\rm The}\;\, {\rm surface}\;\,S\;\,{\rm has}\,\; {\rm pure}\;\, {\rm cohomology}.}\end{align*}

Let $t_1,t_2$ be the Chern roots of $S$ . The Chern classes of $S$ are $c_1=t_1+t_2$ , $c_2=t_1t_2$ . The Todd class is

(2.1) \begin{align} {\rm Td}_S=t_1t_2/(1-e^{-t_1})(1-e^{-t_2}). \end{align}

We may use its graded version

\begin{align*} {\rm Td}_S(z) = z^2t_1t_2/(1-e^{-t_1z})(1-e^{-t_2z}) = \sum _{k{\geqslant } 0} {\rm Td}_S^{(k)} z^k. \end{align*}

Set $s_2=t_1^2+t_1t_2+t_2^2$ . We will also consider the cohomology with compact support $H^*_c(S,\mathbb{Q})$ . Recall that there is an algebra morphism $H^*_c(S,\mathbb{Q}) \to H^*(S,\mathbb{Q})$ and cup product maps $H^i_c(S,\mathbb{Q}) \otimes H^j(S,\mathbb{Q}) \to H^{i+j}_c(S,\mathbb{Q})$ . We set $K^c_0({S})_{\mathbb{Q}} = \bigoplus _i H^{2i}_c(S,\mathbb{Q})$ and denote by

\begin{align*}\langle \alpha ,\beta \rangle =\int _S \alpha ^\vee \cup \beta \cup {\rm Td}_S\end{align*}

the Mukai pairing on $K^c_0({S})_{\mathbb{Q}}$ , where if $\alpha =\sum _k \alpha _k$ with $\alpha _k \in H^{2k}_c(S,\mathbb{Q})$ then $\alpha ^\vee =\sum _k (-1)^k \alpha _k$ .

Taking the generic rank of a coherent sheaf yields a linear map $\operatorname {rk}\nolimits : K^c_0({S})_{\mathbb{Q}} \to \mathbb{Q}$ . The class of the structure sheaf of a point will be denoted by $\delta$ . Note that for any $\alpha$ ,

(2.2) \begin{align} \langle \alpha ,\delta \rangle =\langle \delta , \alpha \rangle =\operatorname {rk}\nolimits (\alpha ). \end{align}

Let us pick some $\alpha \in K^c_0({S})_{\mathbb{Q}}$ . Consider the derived stack $\mathfrak{Coh}_{\alpha }(S)$ parametrizing coherent sheaves $\mathcal{E} \in Coh(S)$ with proper support and Chern character $\alpha$ , see e.g. [Reference Porta and SalaPS23]. Its underlying classical stack will be denoted $\mathcal{C}oh_\alpha (S)$ . Note that the stack $\mathcal{C}oh_\alpha (S)$ parametrizes coherent sheaves on $S$ with proper support and Chern character $\alpha$ , while $Coh(S)$ parametrizes all coherent sheaves on $S$ . When the surface $S$ is understood, we may abbreviate $\mathfrak{Coh}_{\alpha }=\mathfrak{Coh}_{\alpha }(S)$ and likewise for $\mathcal{C}oh_\alpha$ . In addition, when $\alpha =r [\mathcal{O}_{{S}}] + \alpha '$ , where $\alpha ' \in \bigoplus _{i \gt 0} H_c^{2i}(S,\mathbb{Q})$ , we may write $\mathfrak{Coh}_{r,\alpha '}$ instead of $\mathfrak{Coh}_{\alpha }$ . Note that $\mathfrak{Coh}_{\alpha }$ is empty for $\operatorname {rk}\nolimits (\alpha ) \neq 0$ if $S$ is not complete. The stack $\mathcal{C}oh_{\alpha }$ is singular in general, but $\mathfrak{Coh}_{\alpha }$ is quasi-smooth and of virtual dimension $d_{\alpha }=-\langle \alpha , \alpha \rangle$ . Unless $\alpha \in \mathbb{N}\delta$ , the stack $\mathfrak{Coh}_{\alpha }$ is of infinite type. However, it may always be covered by open global quotient stacks which are of finite type. We say that a coherent sheaf $\mathcal{E}$ on $S$ is of dimension ${\geqslant } d$ if it contains no subsheaf with support of dimension strictly less than $d$ . Let $\mathfrak{Coh}_{\alpha }^{{\geqslant } d}$ be the stack parametrizing dimension ${\geqslant } d$ sheaves in $\mathfrak{Coh}_{\alpha }$ . It is open in $\mathfrak{Coh}_{\alpha }$ . We denote by $\mathcal{E}_\alpha \in Coh(\mathfrak{Coh}_\alpha \times S)$ the tautological sheaf. Its restriction to $\mathfrak{Coh}^{{\geqslant } d} \times S$ will be denoted $\mathcal{E}_\alpha ^{{\geqslant } d}$ . As all sheaves in $\mathfrak{Coh}_{\alpha }$ are properly supported, we can extend $\mathcal{E}_\alpha$ by zero and consider it as a sheaf on $\mathfrak{Coh}_\alpha \times \overline {S}$ . An analogous remark holds for $\mathcal{E}_\alpha ^{{\geqslant } d}$ .

2.2 The stack of zero-dimensional sheaves

For any $d \in \mathbb{N}$ , the stack $\mathfrak{Coh}_{d\delta }$ is the derived moduli stack of (zero-dimensional) sheaves on $S$ of length $d$ . Its underlying classical stack $\mathcal{C}oh_{d\delta }$ is irreducible of dimension $d$ while $\mathfrak{Coh}_{d\delta }$ is of virtual dimension $-\langle d\delta ,d\delta \rangle=0$ . Let us set $\mathfrak{Coh}^0=\bigsqcup _d \mathfrak{Coh}_{d\delta }$ and $\mathcal{C}oh^0=\mathfrak{Coh}^{0,cl}$ .

Let $\overline S$ be a smooth compactification of $S$ . Let $\Delta _S : S \to S \times \overline {S}$ be the diagonal map and

\begin{align*}\Delta =\Delta _{S*}(1)\in H^*(S\times {\overline {S}},\mathbb{Q})\end{align*}

be the class of the diagonal. Let $\rho \in Coh(B\mathbb{G}_m)$ be the linear character. We define

(2.3) \begin{align} u=c_1(\rho ). \end{align}

Recall that $ \mathcal{E}_{\delta }\in Coh(S \times \overline {S} \times B\mathbb{G}_m)$ . We have

(2.4) \begin{align} \begin{split} \mathcal{C}oh_{\delta } & \simeq S \times B\mathbb{G}_m, \\ H^*(\mathcal{C}oh_{\delta },\mathbb{Q}) & = H^*(S,\mathbb{Q})[u],\\ \mathcal{E}_{\delta } & =\Delta _{S*}(\mathcal{O}_{S})\boxtimes \rho ,\\ {\rm ch}(\mathcal{E}_{\delta }) & =\Delta \cup e^u\cup {\rm Td}_S^{-1}. \end{split} \end{align}

Since $\mathcal{C}oh_{\delta }$ is smooth, there is an isomorphism

(2.5) \begin{align} H^i(\mathcal{C}oh_{\delta },\mathbb{Q}) = H_{2-i}(\mathcal{C}oh_{\delta },\mathbb{Q}) ,\quad c\mapsto c \cap [\mathcal{C}oh_{\delta }]. \end{align}

We may assume that the surface $S$ is acted upon by a torus $T$ . In this case there is an induced action of $T$ on the stacks $\mathfrak{Coh}_{d\delta }$ , and all homology groups acquire module structure over the ring

\begin{align*}{\bf R}_T=H^*(BT,\mathbb{Q}).\end{align*}

2.3 The COHA of zero-dimensional sheaves

We now introduce, following [Reference Kapranov and VasserotKV23, Section 4], the cohomological Hall algebra of zero-dimensional sheaves on $S$ . See also [Reference ZhaoZha21] for another construction of this COHA, [Reference MinetsMin20] for the case of the cotangent bundle of a curve, and [Reference Schiffmann and VasserotSV13b] for the case of $S=\mathbb{A}^2$ . We consider the $\mathbb{Z}^2$ -graded vector space

\begin{align*} {\bf H}_0(S)= H_*(\mathcal{C}oh^0,\mathbb{Q}),\qquad {\bf H}_0(S)[l,n]=H_{n}(\mathcal{C}oh_{l\delta },\mathbb{Q}). \end{align*}

Let us briefly recall the definition of the COHA product. Fix $\alpha =a\delta , \beta =b\delta$ and $\gamma =\alpha +\beta$ . Let $\widetilde {\mathfrak{Coh}}_{\alpha ;\beta }$ be the derived stack parametrizing short exact sequences $0 \to \mathcal{T}'\to \mathcal{T}\to \mathcal{T}'' \to 0$ with $\mathcal{T}, \mathcal{T}', \mathcal{T}''$ respectively in $\mathfrak{Coh}_\gamma , \mathfrak{Coh}_\beta$ and $\mathfrak{Coh}_\alpha$ . There is a convolution diagram

(2.6) \begin{align} \mathfrak{Coh}_\alpha \times \mathfrak{Coh}_\beta \xleftarrow {\quad q_{\alpha ,\beta }\quad } \widetilde {\mathfrak{Coh}}_{\alpha ;\beta } \xrightarrow {\quad p_{\alpha ,\beta }\quad } \mathfrak{Coh}_\gamma \end{align}

in which the maps $p_{\alpha ,\beta }$ and $q_{\alpha ,\beta }$ assign to the sequence $0 \to \mathcal{T}' \to \mathcal{T} \to \mathcal{T}'' \to 0$ the object $\mathcal{T}$ and the pair of objects $(\mathcal{T}'', \mathcal{T}')$ respectively. The classical truncation of that diagram reads

(2.7) \begin{align} {\mathcal{C}oh_\alpha \times \mathcal{C}oh_\beta \xleftarrow {q_{\alpha ,\beta }^{cl}} \widetilde {\mathcal{C}oh}_{\alpha ;\beta } \xrightarrow {p_{\alpha ,\beta }^{cl}} \mathcal{C}oh_\gamma . } \end{align}

The map $p_{\alpha ,\beta }$ is proper and representable. The map $q_{\alpha ,\beta }$ is neither representable nor smooth, but it is quasi-smooth. More precisely, consider the complex

\begin{align*}\mathcal{C}_{\alpha ,\beta }=\operatorname {RHom}\nolimits _{S}(\mathcal{E}_\alpha , \mathcal{E}_\beta )[1]= {\bf R}p_{12*}{\bf R}\mathcal{H}om(p_{13}^*\mathcal{E}_\alpha , p_{23}^*\mathcal{E}_\beta )[1]\end{align*}

of perfect amplitude $[-1,1]$ . Here $p_{ij}$ stands for the projection from $\mathfrak{Coh}_\alpha \times \mathfrak{Coh}_\beta \times S$ to the $i$ -th and $j$ -th components. There is a canonical isomorphism of derived stacks over $\mathfrak{Coh}_\alpha \times \mathfrak{Coh}_\beta$

\begin{align*}\mathbb{V}(\mathcal{C}_{\alpha ,\beta }) \simeq \widetilde {\mathfrak{Coh}}_{\alpha ,\beta }\end{align*}

where $\mathbb{V}(\mathcal{C}_\bullet )$ is the derived stack given by the total space of a complex $\mathcal{C}_\bullet$ . We may thus define a virtual pullback morphism

\begin{align*}q_{\alpha ,\beta }^!: H_i(\mathcal{C}oh_\alpha \times \mathcal{C}oh_\beta ,\mathbb{Q}) \to H_{i-2\langle \alpha ,\beta \rangle }(\widetilde {\mathcal{C}oh}_{\alpha ;\beta },\mathbb{Q}).\end{align*}

It is useful to rephrase this construction in classical terms. Let us fix an explicit representative of the complex $\mathcal{C}_{\alpha ,\beta }$

\begin{align*}0\to \mathcal{V}_{-1} \to \mathcal{V}_0 \to \mathcal{V}_1\to 0.\end{align*}

Let $\mathcal{C}_{\alpha ,\beta }^{cl}$ be the restriction of $\mathcal{C}_{\alpha ,\beta }$ to $\mathcal{C}oh_\alpha \times \mathcal{C}oh_\beta$ . Let $\tau _{{\leqslant } 0}$ and $(-)^{{\leqslant } 0}$ be the standard and stupid truncations. By [Reference Kapranov and VasserotKV23], there is an isomorphism

\begin{align*}\mathbb{V}(\tau _{{\leqslant } 0}(\mathcal{C}^{cl}_{\alpha ,\beta }))= \widetilde {\mathcal{C}oh}_{\alpha ,\beta }.\end{align*}

This yields a factorization

The map $\pi$ is a linear stack, in particular it is smooth. Hence it yields an isomorphism

\begin{align*}\pi ^*:H_{i}(\mathcal{C}oh_\alpha \times \mathcal{C}oh_\beta ,\mathbb{Q}) \to H_{i+2d_0}(\mathbb{V}(\mathcal{C}^{cl,{\leqslant } 0}_{\alpha ,\beta }),\mathbb{Q}).\end{align*}

Further, there is a refined Gysin pullback

\begin{align*}\iota ^! : H_i(\mathbb{V}(\mathcal{C}^{cl,{\leqslant } 0}_{\alpha ,\beta }),\mathbb{Q}) \to H_{i-2d_1}(\widetilde {\mathcal{C}oh}_{\alpha ,\beta },\mathbb{Q}).\end{align*}

Here $d_0$ and $d_1$ are the ranks of $\mathcal{C}^{{\leqslant } 0}_{\alpha ,\beta }$ and $\mathcal{V}_1$ , so $d_0-d_1=-\langle \alpha ,\beta \rangle$ is the virtual rank of $\mathcal{C}^{cl}_{\alpha ,\beta }$ . We have $q_{\alpha ,\beta }^!= \iota ^!\circ \pi ^*$ , see [Reference KhanKha19]. In particular, the morphism $\iota ^!\circ \pi ^*$ thus defined is independent of the presentation of the complex $\mathcal{C}$ . See [Reference Kapranov and VasserotKV23, Section 3] for a direct proof. We set

\begin{align*} \begin{split} \star = (p_{\alpha ,\beta })_*\circ q^!_{\alpha ,\beta } : H_*(\mathcal{C}oh_\alpha ,\mathbb{Q}) \otimes H_*(\mathcal{C}oh_\beta ,\mathbb{Q}) & = H_*(\mathcal{C}oh_\alpha \times \mathcal{C}oh_\beta ,\mathbb{Q})\\ & \to H_{* -2\langle \alpha ,\beta \rangle }(\mathcal{C}oh_{\gamma },\mathbb{Q}). \end{split} \end{align*}

In the presence of a torus $T$ , we may consider the $T$ -equivariant COHA which is an algebra with underlying vector space given by

\begin{align*} {\bf H}_0^T(S)=H^T_*(\mathcal{C}oh^0,\mathbb{Q}). \end{align*}

An open inclusion $i:S \to S'$ of smooth surfaces gives an open inclusion of derived stacks $\underline {i}:\mathfrak{Coh}^0(S) \to \mathfrak{Coh}^0(S')$ and thus a map $\underline {i}^* : {\bf H}_0(S') \to {\bf H}_0(S)$ .

We will need the following result on the Hilbert series of ${\bf H}_0(S)$ . We define

\begin{align*}h_{{\bf H}_0(S)}(z,w)=\sum _{l,n} \operatorname {dim}\nolimits ({\bf H}_0(S)[l,n])(-z)^nw^l.\end{align*}

Let $h_S(z)=\sum _n \operatorname {dim}\nolimits (H_n(S,\mathbb{Q})) (-z)^n$ be the Borel-Moore homology Poincaré polynomial of $S$ .

Theorem 2.5 ([Reference Kapranov and VasserotKV23, Theorem 7.1.6]). Let $q,t$ be formal variables of respective degrees $[0,-2]$ and $[1,0]$ . There is a canonical isomorphism of graded vector spaces

\begin{align*}{\bf H}_0(S) = \operatorname {Sym}\nolimits \left (H_*(S \times B\mathbb{G}_m,\mathbb{Q}) \otimes qt\mathbb{Q}[t])\right )=\operatorname {Sym}\nolimits \left (H_*(S,\mathbb{Q}) \otimes qt\mathbb{Q}[q,t]\right )\!.\end{align*}

In particular, the Hilbert series of ${\bf H}_0(S)$ is given by

\begin{align*}h_{{\bf H}_0(S)}(z,w)= \operatorname {Exp}\nolimits \left ( \frac {h_S(z)z^{-2}w}{(1-z^{-2})(1-w)}\right )\end{align*}

where $\operatorname {Exp}\nolimits$ is the plethystic exponential.

Theorem 2.5 is proved for an arbitrary smooth surface $S$ using factorization homology techniques, which do not extend to the $T$ -equivariant setting. However, in [Reference Davison, Hennecart and Schlegel MejiaDHS23] the question of equivariant formality is treated in the much greater generality of relative COHAs, which includes our COHAs by [Reference Davison, Hennecart and Schlegel MejiaDHS22, Section 11.1]. In particular, as soon as $S$ is $T$ -equivariantly formal, ${\bf H}_0(S)$ is a free ${\bf R}_T$ -module of (graded) rank given by $h_{{\bf H}_0(S)}(z,w)$ by [Reference Davison, Hennecart and Schlegel MejiaDHS23, Theorems 11.5, 11.6]. By [Reference Goresky, Kottwitz and MacPhersonGKM98, Theorem 14.1] this assumption is satisfied for $S$ cohomologically pure.

2.4 The compactly supported COHA of zero dimensional sheaves

When $S$ is not proper, we will also consider a variant of ${\bf H}_0(S)$ defined using the hyperbolic Borel-Moore homology introduced in Appendix A.1. More precisely, set $\operatorname {Sym}\nolimits (S)=\bigsqcup _n \operatorname {Sym}\nolimits ^n(S)$ . Let

\begin{align*}\mathfrak{Coh}^0 \xrightarrow {{\rm supp}} \operatorname {Sym}\nolimits (S)\xrightarrow {\quad \pi \quad }{\rm pt}\end{align*}

be the support and the projection to a point. We set

\begin{gather*} {\bf H}_0^{c}(S)=\bigoplus _d H^c_*(\mathcal{C}oh_{d\delta },\mathbb{Q}), \end{gather*}

where $H^c_*(\mathcal{C}oh_{d\delta },\mathbb{Q})=H_*(\mathfrak{Coh}_{d\delta }/\operatorname {Sym}\nolimits ^d(S),\mathbb{Q})$ and $H_*(\mathfrak{Coh}_{d\delta }/\operatorname {Sym}\nolimits ^d(S),\mathbb{Q})$ is defined by

\begin{gather*} H_*(\mathfrak{Coh}_{d\delta }/\operatorname {Sym}\nolimits ^d(S),\mathbb{Q})=H^{-*}(\pi _!{\rm supp}_*\mathbb{D}_{\mathfrak{Coh}_{d\delta }}). \end{gather*}

The map $H^*_c(S,\mathbb{Q})[u] \to H^c_{*}(\mathcal{C}oh_\delta ,\mathbb{Q})$ given by $x \mapsto x \cap [\mathcal{C}oh_\delta ]$ is an isomorphism. We complete the induction diagram by introducing the support maps

(2.8)

where $\oplus$ is the direct sum (a finite map), which allows us to view both $\mathfrak{Coh}_{m\delta } \times \mathfrak{Coh}_{n\delta }$ and $\widetilde {\mathfrak{Coh}}_{m\delta ;n\delta }$ as derived stacks over $\operatorname {Sym}\nolimits ^{m+n}(S)$ . Note that

\begin{align*} \begin{split} H_*(\mathcal{C}oh_{m\delta } & \times \mathcal{C}oh_{n\delta } / \operatorname {Sym}\nolimits ^{m+n}(S),\mathbb{Q}) \\ & = H_*(\mathcal{C}oh_{m\delta } \times \mathcal{C}oh_{n\delta } / \operatorname {Sym}\nolimits ^{m}(S)\times \operatorname {Sym}\nolimits ^{n}(S),\mathbb{Q})\\ & = H_*(\mathcal{C}oh_{m\delta }/\operatorname {Sym}\nolimits ^m(S),\mathbb{Q})\otimes H_*(\mathcal{C}oh_{n\delta }/\operatorname {Sym}\nolimits ^n(S),\mathbb{Q}); \end{split} \end{align*}

here the first equality follows from (A.1) and properness of $\oplus$ . We may now define the convolution product

\begin{align*} \begin{split} \star = (p_{m,n})_!\circ q_{m,n}^*: H^c_*(\mathcal{C}oh_{m\delta },\mathbb{Q}) \otimes H^c_*(\mathcal{C}oh_{n\delta },\mathbb{Q}) & = H^c_*(\mathcal{C}oh_{m\delta }\times \mathcal{C}oh_{n\delta },\mathbb{Q})\\ & \to H^c_*(\mathcal{C}oh_{(m+n)\delta },\mathbb{Q}). \end{split} \end{align*}

If the surface $S$ is projective, then we have ${\bf H}_0^c(S)={\bf H}_0(S)$ . For each open embedding $i: S \to S'$ we have an open embedding of derived stacks $\underline {i}:\mathfrak{Coh}_{n\delta }\to \mathfrak{Coh}_{n\delta }(S')$ . Hence, there are pushforward maps $\underline {i}_!:H_*^c(\mathfrak{Coh}_{n\delta },\mathbb{Q}) \to H_*^c(\mathfrak{Coh}_{n\delta }(S'),\mathbb{Q})$ , which combine to $\underline {i}_!:{\bf H}_0^c(S) \to {\bf H}^c_0(S')$ .

If $\iota : S \to {\overline {S}}$ is a smooth compactification of $S$ , Lemmas 2.4 and 2.7 yield algebra homomorphisms

(2.9) \begin{align} {{\bf H}_0^c(S) \xrightarrow {\,\,\underline {\iota }_!\,\,} {\bf H}^c_0({\overline {S}}) = {\bf H}_0({\overline {S}}) \xrightarrow {\,\,\underline {\iota }^*\,\,} {\bf H}_0(S).} \end{align}

The composed map

\begin{align*}\phi _S=\underline {\iota }^*\underline {\iota }_! : {\bf H}_0^c(S) \to {\bf H}_0(S)\end{align*}

is independent of the choice of $\overline {S}$ . Let

\begin{align*}h_S^c(z)=\sum _n \operatorname {dim}\nolimits (H_n^c(S,\mathbb{Q})) (-z)^n\end{align*}

be the homology Poincaré polynomial of $S$ . We will need the following variant of Theorem 2.5, which can be found in [Reference Davison, Hennecart and Schlegel MejiaDHS23, Corollary 7.11].

Theorem 2.8. There is a canonical isomorphism of graded vector spaces

\begin{align*}{\bf H}^c_0(S) = \operatorname {Sym}\nolimits \left (H^c_*(S \times B\mathbb{G}_m,\mathbb{Q}) \otimes qt\mathbb{Q}[t])\right )=\operatorname {Sym}\nolimits \left (H^c_*(S,\mathbb{Q}) \otimes qt\mathbb{Q}[q,t]\right )\!.\end{align*}

In particular, the Hilbert series of ${\bf H}^c_0(S)$ is given by

\begin{align*}h_{{\bf H}^c_0(S)}(z,w)=\operatorname {Exp}\nolimits \left (\frac {h^c_S(z)z^{-2}w}{(1-z^{-2})(1-w)}\right )\!.\end{align*}

As explained after Theorem 2.5, a $T$ -equivariant version of Theorem 2.8 follows from [Reference Davison, Hennecart and Schlegel MejiaDHS23, Section 11] provided that $S$ is pure.

2.5 The COHA of properly supported sheaves

Following [Reference Kapranov and VasserotKV23, Section 4], one can extend the construction of COHA product to the stack of properly supported sheaves on $S$ . We set

\begin{align*}{\bf H}(S)= \bigoplus _{\alpha } H_*(\mathcal{C}oh_{\alpha },\mathbb{Q}).\end{align*}

As in the case of zero-dimensional sheaves, for any $\alpha ,\beta \in K^c_0(S)_{\mathbb{Q}}$ there is an induction diagram

(2.10) \begin{align} \mathfrak{Coh}_\alpha \times \mathfrak{Coh}_\beta \xleftarrow {q_{\alpha ,\beta }} \widetilde {\mathfrak{Coh}}_{\alpha ;\beta } \xrightarrow {p_{\alpha ,\beta }}\mathfrak{Coh}_\gamma ,\quad \gamma =\alpha +\beta \end{align}

with $q_{\alpha ,\beta }$ quasi-smooth and $p_{\alpha ,\beta }$ proper.

Assume that $i: S \to S'$ is an open immersion into another smooth surface $S'$ . This gives rise to an open immersion $\underline {i}:\mathfrak{Coh}(S) \to \mathfrak{Coh}(S')$ and hence to a restriction morphism $\underline {i}^* : {\bf H}(S') \to {\bf H}(S)$ . The proof of Lemma 2.4 implies that the map $\underline {i}^*$ is an algebra homomorphism.

2.6 Tautological classes

Let us now study a family of cohomology classes $ch_i(\lambda )$ on each $\mathcal{C}oh_{\alpha }$ for $i {\geqslant } 1$ and $\lambda \in H^*(S,\mathbb{Q})$ . Fix a smooth compactification $\iota : S \to \overline {S}$ of $S$ . Since $S$ is pure, the restriction $H^*(\overline {S},\mathbb{Q})\to H^*(S,\mathbb{Q})$ is surjective. Let $I(S)\subset H^*(\overline {S},\mathbb{Q})$ be the kernel of this map. It is identified with the relative cohomology $H^*({\overline {S}},S,\mathbb{Q})$ . The map pushforward $H^*_c(S,\mathbb{Q})\to H^*(\overline {S},\mathbb{Q})$ is injective. The intersection pairing on $H^*(\overline {S},\mathbb{Q})$ identifies $H^*_c(S,\mathbb{Q})$ with the orthogonal complement $I(S)^\perp$ in $H^*(\overline S,\mathbb{Q})$ . This is a (non-unital) subalgebra under the cup product. Dually, $H^*(S,\mathbb{Q})$ and $H^*(\overline {S},\mathbb{Q})$ are equipped with natural coproducts and the restriction $\iota ^*$ is a coalgebra homomorphism.

When $S$ is not proper, it will be convenient to formally add a class $[{\rm pt}]$ of degree $4$ to $H^*(S,\mathbb{Q})$ , satisfying $[{\rm pt}] \cup H^{\gt 0}(S,\mathbb{Q})=\{0\}$ . Likewise, it will be convenient to formally add a unit $1$ to $H^*_c(S,\mathbb{Q})$ . We will modify the coproduct accordingly. Let

\begin{align*}\overline {H}^*(S,\mathbb{Q}),\quad \overline {H}^*_c(S,\mathbb{Q})\end{align*}

be the resulting rings. If $\Delta '$ is the coproduct on $H^*(S,\mathbb{Q})$ we define the coproduct of $\overline {H}^*(S,\mathbb{Q})$ by

\begin{align*} \Delta : \overline {H}^*(S,\mathbb{Q}) \to \overline {H}^*(S,\mathbb{Q}) \otimes \overline {H}^*(S,\mathbb{Q}), \end{align*}

such that

\begin{align*} \Delta ([{\rm pt}])=[{\rm pt}] \otimes [{\rm pt}],\quad \Delta (\lambda )=[{\rm pt}] \otimes \lambda + \lambda \otimes [{\rm pt}] + \Delta '(\lambda ),\quad \lambda \in H^*(S,\mathbb{Q}). \end{align*}

Let us define a variant of the ring of symmetric function which is colored by $H^*(S,\mathbb{Q})$ . Consider

\begin{align*} U(S)=\operatorname {Sym}\nolimits (H^*(S,\mathbb{Q})\otimes t\mathbb{Q}[t]). \end{align*}

For each $i {\geqslant } 1$ and $\lambda \in H^*(S,\mathbb{Q})$ , we denote $\underline {{\rm ch}}_i(\lambda ) = \lambda \otimes t^i$ . Set ${\rm deg}(\underline {{\rm ch}}_i(\lambda ))=2i + {\rm deg}(\lambda )-4$ . To keep track of the rank of coherent sheaves, we add an extra element ${\bf r}$ of degree $0$ and set

\begin{align*}U'(S) = U(S) \otimes \mathbb{Q}[{\bf r}].\end{align*}

It is the free graded-commutative algebra generated by $\underline {{\rm ch}}_i(\lambda )$ and ${\bf r}$ subject to the relations

\begin{align*}\underline {{\rm ch}}_i(\lambda +\mu )=\underline {{\rm ch}}_i(\lambda ) + \underline {{\rm ch}}_i(\mu ), \quad \underline {{\rm ch}}_i(a\lambda )=a\,\underline {{\rm ch}}_i(\lambda ),\end{align*}

for any $a \in \mathbb{Q}$ and $\lambda ,\mu \in H^*(S,\mathbb{Q})$ .

Definition 2.11. Let $\Lambda (S)$ be the quotient of $U'(S)$ by the ideal generated by the negative degree elements $\underline {{\rm ch}}_1(\lambda )$ for $\deg (\lambda )=0,1$ . The universal Chern character $\underline {{\rm ch}}(z)$ is defined as follows:

\begin{align*} \underline {{\rm ch}}(z)={\bf r} \otimes 1 +\sum _{i{\geqslant } 1} \underline {{\rm ch}}_{i} z^i \in \Lambda (S)\otimes \overline {H}^*_c(S,\mathbb{Q})[[z]], \qquad \underline {{\rm ch}}_i=\sum _\lambda \underline {{\rm ch}}_{i}(\lambda )\otimes \lambda ^*, \end{align*}

where $\sum \lambda \otimes \lambda ^*\in H^*(S,\mathbb{Q}) \otimes H^*_c(S,\mathbb{Q})$ is the intersection pairing tensor.

We define a coalgebra structure on $\Lambda (S)$ by requiring the elements $\underline {{\rm ch}}_i(\lambda )$ and ${\bf r}$ to be primitive. In other words,

\begin{align*}(\Delta \otimes {\rm Id})(\underline {{\rm ch}}(z))=\underline {{\rm ch}}(z)_{13} + \underline {{\rm ch}}(z)_{23} \in \Lambda (S) \otimes \Lambda (S) \otimes \overline {H}^*_c(S,\mathbb{Q})[[z]].\end{align*}

We also consider an involution

(2.11) \begin{align} \upsilon : \Lambda (S) \to \Lambda (S), \quad \upsilon ({\bf r})=-{\bf r}, \quad \upsilon (\underline {{\rm ch}}_i(\lambda ))=-\underline {{\rm ch}}_i(\lambda ), \quad i {\geqslant } 1,\quad \lambda \in H^*(S,\mathbb{Q}). \end{align}

The elements $\underline {{\rm ch}}_i$ , being even, commute with each other. Recall the algebra $\Lambda '$ from (1.4). We may define an algebra morphism

\begin{align*}p: \Lambda ' \to \Lambda (S) \otimes \overline {H}^*_c(S,\mathbb{Q}), \quad p_0 \mapsto {\bf r} \otimes 1, \qquad p_i/i! \mapsto \underline {{\rm ch}}_i,\quad i {\geqslant } 1.\end{align*}

We will use the following notation

(2.12) \begin{align} f(\lambda )=\int _S p(f) \cup \lambda \in \Lambda (S),\quad f \in \Lambda ',\quad \lambda \in \overline {H}^*(S,\mathbb{Q}), \end{align}

where the map

\begin{align*}\int _S:\overline H^*_c(S,\mathbb{Q})\to \mathbb{Q}\end{align*}

is defined in the obvious way. We may simply write $f(\lambda )=\int _S f \lambda$ when there is no risk of confusion. For instance, we have $p_i(\lambda )=i! \underline {{\rm ch}}_i(\lambda )$ . Observe that

(2.13) \begin{align} (f \cdot g)(\lambda )=(f \otimes g)(\Delta (\lambda ))=\sum f(\lambda ^{(1)})g(\lambda ^{(2)}) \end{align}

for $f,g \in \Lambda '$ and $\lambda \in \overline {H}^*(S,\mathbb{Q})$ , where $\Delta (\lambda )=\sum \lambda ^{(1)} \otimes \lambda ^{(2)}$ in Sweedler’s notation. In particular,

\begin{align*}(p_{i_1} \cdots p_{i_l})(\lambda )=i_1! \cdots i_l!\sum \underline {{\rm ch}}_{i_1}(\lambda ^{(1)}) \cdots \underline {{\rm ch}}_{i_l}(\lambda ^{(l)}).\end{align*}

We have ${\rm deg}(f(\lambda ))=2{\rm deg}(f) + \deg (\lambda )-4$ .

Remark 2.15. Note that with our conventions we have

\begin{align*}h_0([{\rm pt}])=1([{\rm pt}])=1 \in \Lambda (S), \qquad p_0([{\rm pt}]) = {\bf r} \otimes 1([{\rm pt}]) ={\bf r} \in \Lambda (S)\end{align*}

while $1(\lambda )=p_0(\lambda )=0$ if $\deg \lambda \lt 4$ , regardless of whether $S$ is proper or not. In a similar vein, we have $f([{\rm pt}])=0$ for any $f$ in the augmentation ideal of $\mathbb{Q}[p_1, p_2, \ldots ]$ when $S$ is not proper.

Now fix $\alpha \in K_0^c({\overline {S}})_{\mathbb{Q}}$ a class of rank $r$ and consider a locally closed substack $\mathcal{U}_\alpha \subset \mathcal{C}oh_\alpha ({\overline {S}})$ . Let $\mathcal{E}_{\alpha } \in Coh(\mathcal{U}_{\alpha }\times \overline {S})$ denote the restriction of the tautological sheaf to $\mathcal{C}oh_{\alpha }(\overline {S}) \times \overline {S}$ . Consider its Chern character

\begin{align*} {\rm ch}(\mathcal{E}_\alpha ,z) = r + \sum _{i{\geqslant } 1}p_i(\mathcal{E}_\alpha )z^i/i! \end{align*}

in

\begin{align*}H^*(\mathcal{U}_{\alpha }\times \overline {S},\mathbb{Q})=H^*(\mathcal{U}_{\alpha },\mathbb{Q})\otimes H^*(\overline {S},\mathbb{Q})\end{align*}

where $p_i(\mathcal{E}_\alpha )$ is as in (1.6). We have a unique graded ring homomorphism

(2.14) \begin{align} \operatorname {ev}\nolimits _{\alpha } : \Lambda ({{\overline {S}}}) \to H^*(\mathcal{U}_{\alpha },\mathbb{Q}), \end{align}

defined by

\begin{align*}(\operatorname {ev}\nolimits _{\alpha } \otimes {\rm Id})(\underline {{\rm ch}}(z))={\rm ch}(\mathcal{E}_{\alpha },z) \in H^*(\mathcal{U}_{\alpha },\mathbb{Q}) \otimes H^*({\overline {S}},\mathbb{Q})[[z]].\end{align*}

Observe that $r = \operatorname {ev}\nolimits _\alpha ({\bf r})$ . The following lemma is a straightforward corollary of Remark 2.12.

The condition of Lemma 2.16 is verified if $\mathcal{U}_\alpha \subset \mathcal{C}oh_{\alpha }$ . It also holds if the restriction of $\mathcal{E}_\alpha$ to $\mathcal{U}_\alpha \times ({\overline {S}}\setminus S)$ is a trivial vector bundle. Such situations occur when considering moduli stacks of sheaves on $\overline {S}$ which are trivialized along ${\overline {S}}\setminus S$ . From now on, unless specified otherwise, we will assume that $\mathcal{U}_\alpha =\mathcal{C}oh_\alpha$ . Then, the map $\operatorname {ev}\nolimits _{\alpha }$ is a ring homomorphism $\Lambda ({{\overline {S}}}) \to H^*(\mathcal{C}oh_{\alpha },\mathbb{Q})$ which factors through $\Lambda (S) \to H^*(\mathcal{C}oh_{\alpha },\mathbb{Q}).$

2.7 Extended COHAs

Assume now that $\alpha \in \mathbb{N} \delta$ . Composing the map $\operatorname {ev}\nolimits _{\alpha }$ above with the cap product yields an action $\bullet$ of $\Lambda ({S})$ on ${\bf H}_0(S)$ such that

\begin{align*} x \bullet c=\operatorname {ev}\nolimits _{\alpha }(x) \cap c ,\quad c \in H_*(\mathcal{C}oh_{\alpha },\mathbb{Q}) ,\quad x \in \Lambda ({S}). \end{align*}

This action preserves each $H_*(\mathcal{C}oh_{\alpha },\mathbb{Q})$ and is compatible with the (co)homological gradings, i.e. ${\rm deg}(x\bullet c)={\rm deg}(c)-{\rm deg}(x)$ . The same holds for ${\bf H}^c_0(S)$ , where the action of $H^*(\mathcal{C}oh_\alpha ,\mathbb{Q})$ on $H^c_*(\mathcal{C}oh_\alpha ,\mathbb{Q})$ is given by Lemma A.3.

Example 2.17. Assuming that $S$ is proper, let us compute $\operatorname {ev}\nolimits _{n\delta }(p_1([{\rm pt}]))$ . By definition, we have

\begin{align*}\sum _\lambda \operatorname {ev}\nolimits _{n\delta }(p_1(\lambda ))\otimes \lambda ^*= c_1(\mathcal{E}_{n\delta }).\end{align*}

Hence $\operatorname {ev}\nolimits _{n\delta }(p_1([{\rm pt}]))=c_1(i_x^*(\mathcal{E}_{n\delta }))$ in $H^2(\mathcal{C}oh_{n\delta },\mathbb{Q})$ for any closed point $i_x: \operatorname {Spec}\nolimits (\mathbb{C}) \to S$ . The support of $i_x^*(\mathcal{E}_{n\delta })$ being of codimension $2$ , its first Chern class vanishes, hence $\operatorname {ev}\nolimits _{n\delta }(p_1([{\rm pt}]))=0$ for any $n\gt 0$ .

Proposition 2.18. The ring ${\bf H}_0(S)$ is a $\Lambda ({S})$ -module algebra, i.e. we have

(2.15) \begin{align} x \bullet (c_1 \star c_2)=\sum (-1)^{|c_1|\cdot |x^{(2)}_i|}(x^{(1)}_i \bullet c_1) \star (x^{(2)}_i \bullet c_2) ,\quad x \in \Lambda (S) ,\quad c_1,c_2 \in {\bf H}_0(S), \end{align}

where $\Delta (x) = \sum x^{(1)}_i\otimes x^{(2)}_i$ . The same holds for ${\bf H}^c_0(S)$ . The map $\phi _S: {\bf H}^c_0(S) \to {\bf H}_0(S)$ is a morphism of $\Lambda (S)$ -modules.

Proof. We will deal with the case of ${\bf H}_0(S)$ , the other one is similar. We can assume that $c_i \in H_*(\mathcal{C}oh_{\alpha _i},\mathbb{Q})$ for $i=1,2$ . Set $\gamma =\alpha _1+\alpha _2$ . Recall the induction diagram (2.6). We abbreviate $p=p_{\alpha _1,\alpha _2}$ and $q=q_{\alpha _1,\alpha _2}$ . By the projection formula

\begin{align*}x \bullet (c_1 \star c_2)=\operatorname {ev}\nolimits _{\gamma }(x) \cap p_*q^!(c_1 \otimes c_2)=p_*(p^*(\operatorname {ev}\nolimits _{\gamma }(x) ) \cap q^!(c_1 \otimes c_2)).\end{align*}

There is a short exact sequence of tautological sheaves

\begin{align*}0 \to q^*(\mathcal{E}_{\alpha _2}) \to p^*(\mathcal{E}_{\gamma }) \to q^*(\mathcal{E}_{\alpha _1})\to 0\end{align*}

Hence $p^*({\rm ch}(\mathcal{E}_{\gamma }))=q^*({\rm ch}(\mathcal{E}_{\alpha _1}) + {\rm ch}(\mathcal{E}_{\alpha _2}))$ . We deduce that

\begin{align*} \begin{split} p_*(p^*(\operatorname {ev}\nolimits _{\gamma }(x) ) \cap q^!(c_1 \otimes c_2)) & =p_*q^!\left ( (\operatorname {ev}\nolimits _{\alpha _1} \otimes \operatorname {ev}\nolimits _{\alpha _2})(\Delta (x)) \cap (c_1 \otimes c_2)\right )\\ & =\sum (-1)^{|c_1|\cdot |x^{(2)}_i|} (x^{(1)}_i \bullet c_1) \star (x^{(2)}_i \bullet c_2). \end{split} \end{align*}

Note that we used Proposition A.1(c) in the first line. The compatibility between $\phi _S$ and the action of $\Lambda (S)$ results from the projection formula in hyperbolic or Borel-Moore homology.

We define the semi-direct product

(2.16) \begin{align} \widetilde {{\bf H}}_0(S)={\bf H}_0(S) \rtimes \Lambda (S) \end{align}

to be the algebra generated by ${\bf H}_0(S)$ and $\Lambda (S)$ modulo the relations

(2.17) \begin{align} x \cdot c=\sum (-1)^{|c|\cdot |x^{(2)}_i|} (x^{(1)}_i \bullet c) \cdot x^{(2)}_i ,\quad x \in \Lambda (S),\; c \in {\bf H}_0(S). \end{align}

The multiplication map $\Lambda (S) \otimes {\bf H}_0(S) \to \widetilde {{\bf H}}_0(S)$ is an isomorphism of graded vector spaces. We define similarly the semi-direct product

(2.18) \begin{align} \widetilde {{\bf H}}^c_0(S)={\bf H}^c_0(S) \rtimes \Lambda (S). \end{align}

We finish with the following observation. The degree one piece of ${\bf H}_0(S)$ is

\begin{align*}{\bf H}_0(S)[1,-]=H_*(\mathcal{C}oh_\delta ,\mathbb{Q})=H_*(S \times B\mathbb{G}_m,\mathbb{Q})=H_*(S,\mathbb{Q})[u].\end{align*}

We consider the linear map

(2.19) \begin{align} \omega _\delta : \Lambda ({S}) \to H_*(\mathcal{C}oh_\delta ,\mathbb{Q}) ,\quad x \mapsto x \bullet [\mathcal{C}oh_\delta ] \end{align}

Proof. The restriction $H_*(\overline {S},\mathbb{Q}) \to H_*(S,\mathbb{Q})$ is surjective. So the restriction $H_*(\mathcal{C}oh_\delta (\overline {S}),\mathbb{Q}) \to H_*(\mathcal{C}oh_\delta ,\mathbb{Q})$ is also surjective. As the latter is a morphism of $\Lambda (\overline {S})$ -module, it is enough to prove the statement for $S$ projective. By (2.4), we have

\begin{align*}{\rm ch}(\mathcal{E}_\delta )={\rm Td}_S^{-1}e^u\Delta _S={\rm Td}_S^{-1}e^u \sum _{\lambda } \lambda \otimes \lambda ^*\end{align*}

where $\{\lambda \},$ $\{\lambda ^*\}$ are dual bases of $H^*(S,\mathbb{Q})$ and $H^*_c(S,\mathbb{Q})$ . Since ${\rm Td}_S$ is invertible, the result follows.

When $S$ is not pure, the map $\operatorname {ev}\nolimits _\delta$ may still be defined but it cannot be surjective since $\mathcal{E}_\delta$ extends to the compactification $\mathcal{C}oh_\delta (\overline {S}) \times \overline {S}$ .

3.1 Hecke correspondences

From (2.10), we can derive the following induction diagrams

(3.1) \begin{align} \begin{gathered} \mathfrak{Coh}_{n\delta } \times \mathfrak{Coh}_\alpha \xleftarrow {q_{n\delta ,\alpha }} \widetilde {\mathfrak{Coh}}_{n\delta ;\alpha } \xrightarrow {p_{n\delta ,\alpha }}\mathfrak{Coh}_{\alpha +n\delta },\\ \mathfrak{Coh}_{\alpha } \times \mathfrak{Coh}_{n\delta }\xleftarrow {\overline {q}_{\alpha ,n\delta }} \widetilde {\mathfrak{Coh}}_{n\delta ;\alpha -n\delta }\xrightarrow {\overline {p}_{\alpha ,n\delta }} \mathfrak{Coh}_{\alpha -n\delta }. \end{gathered} \end{align}

For the compactly supported COHA it is useful to factor the maps $p_{n\delta ,\alpha },$ $ \overline {p}_{\alpha ,n\delta }$ as follows

(3.2) \begin{align} \begin{gathered} \widetilde {\mathfrak{Coh}}_{n\delta ;\alpha }\xrightarrow {p'_{n\delta ,\alpha }} \mathfrak{Coh}_{\alpha +n\delta } \times \operatorname {Sym}\nolimits ^n(S)\xrightarrow {p''_{n\delta ,\alpha }} \mathfrak{Coh}_{\alpha +n\delta },\\ \widetilde {\mathfrak{Coh}}_{n\delta ;\alpha -n\delta } \xrightarrow {\overline {p}'_{\alpha ,n\delta }} \mathfrak{Coh}_{\alpha -n\delta } \times \operatorname {Sym}\nolimits ^n(S) \xrightarrow {\overline {p}''_{\alpha ,n\delta }} \mathfrak{Coh}_{\alpha -n\delta }. \end{gathered} \end{align}

We call the first/second diagram in (3.1) the positive/negative length $n$ Hecke correspondences. Since $\mathfrak{Coh}^{{\geqslant } d}$ is stable under taking subobjects, the Hecke correspondences restrict to $\mathfrak{Coh}^{{\geqslant } d}$ , yielding the following restricted induction diagrams

(3.3) \begin{align} \begin{gathered} {\mathfrak{Coh}_{n\delta } \times \mathfrak{Coh}_\alpha ^{{\geqslant } d} \xleftarrow {q_{n\delta ,\alpha }} \widetilde {\mathfrak{Coh}}_{n\delta ;\alpha }^{{\geqslant } d} \xrightarrow {p_{n\delta ,\alpha }} \mathfrak{Coh}_{\alpha +n\delta }^{{\geqslant } d}, }\\ {\mathfrak{Coh}_{\alpha }^{{\geqslant } d} \times \mathfrak{Coh}_{n\delta } \xleftarrow {\overline {q}_{\alpha ,n\delta }} \widetilde {\mathfrak{Coh}}_{n\delta ;\alpha -n\delta }^{{\geqslant } d} \xrightarrow {\overline {p}_{\alpha ,n\delta }} \mathfrak{Coh}_{\alpha -n\delta }^{{\geqslant } d}, } \end{gathered} \end{align}

where

\begin{align*}\widetilde {\mathfrak{Coh}}^{{\geqslant } d}_{\alpha ;\beta }=\widetilde {\mathfrak{Coh}}_{\alpha ;\beta } \underset {\mathfrak{Coh}_{\alpha +\beta }}{\times } \mathfrak{Coh}^{{\geqslant } d}_{\alpha +\beta }\end{align*}

is the derived fiber product.

3.2 Locally free resolutions

The Hecke correspondences enjoy much better properties when the tautological sheaf $\mathcal{E}$ has perfect amplitude in $[-1,0]$ and admits locally a two-step locally free resolution. This is true in our situation after we restrict to the open substack $\mathfrak{Coh}^{{\geqslant } 1}$ .

Proof. Having a $2$ -step resolution by locally free sheaves is a local condition on any stack of finite type. Indeed, let $\mathcal{F} \in Coh(X)$ , where $X$ is a stack of finite type, and assume that $\mathcal{F}|_{U_i}$ admits $2$ -step resolutions by locally free sheaves for some open cover $\bigcup _i U_i = X$ . Since $X$ is of finite type, there exists a short exact sequence

\begin{align*}0 \to \mathcal{E}_{-1} \to \mathcal{E}_0 \to \mathcal{F}\to 0\end{align*}

with $\mathcal{E}_0$ being a locally free sheaf (see [Reference Huybrechts and LehnHL10, Proposition 2.1.10]). We claim that $\mathcal{E}_{-1}$ is locally free. This may be done locally and on any atlas $[U/R] \to X$ ; it is enough to check that the stalk $(\mathcal{E}_{-1})_{|U,u}$ is a flat $\mathcal{O}_{U,u}$ -module for any $u \in U$ ; this follows from the fact that ${\rm Tor}^{\gt 1}(\mathcal{F}_{|U},-)=\{0\}$ and the standard long exact sequence. Returning to the setting of the lemma, as $\mathcal{E}_{\gamma } |_{ \mathfrak{U}\times \overline S}$ is $\mathfrak{U}$ -flat, it is enough to check that for any $\mathbb{C}$ -point $x\in \mathfrak{U}(\mathbb{C})$ the sheaf $\mathcal{E}_{\gamma }|_{\{x\}\times \overline S}$ admits a $2$ -step resolution by locally free sheaves over $\overline S$ , see [Reference Huybrechts and LehnHL10, Lemma 2.1.7]. This in turn follows from the fact that $\mathcal{E}_{\gamma }|_{\{x\}\times \overline S}$ is of dimension ${\geqslant } 1$ and that $\overline {S}$ is smooth, so that $\mathcal{E}_{\gamma }|_{\{x\}\times \overline S}$ has perfect (equivalently, Tor) amplitude in $[-1,0]$ (see e.g. [Reference Huybrechts and LehnHL10, Part II., 5.]).

In the remainder of this section, we describe the length one Hecke correspondences and compute their action on tautological classes. Until Section 3.4, we let $S$ be an arbitrary smooth connected surface. A general framework for derived Hecke correspondences has recently been worked out by Q. Jiang in [Reference JiangJia25, Section 8], in the language of derived algebraic geometry, following the work of Negut [Reference NegutNeg22].

3.3 Length one Hecke correspondences and operators

Fix $\alpha$ and set $\gamma =\alpha +\delta$ . In this section we consider length one Hecke correspondences given by the diagrams in (3.1) with $n=1$ restricted to $\mathfrak{Coh}^{{\geqslant } 1}$ . To unburden the notation, we will drop the indices of the maps $p$ , $q$ , etc. Let $K_{S}$ be the canonical bundle of $S$ and set $\mathcal{F}_\alpha =\mathcal{E}_{\alpha }^\vee \otimes K_{S}[1]$ , a complex over $\mathfrak{Coh}_\alpha \times S$ . By (2.4), the Serre duality gives an isomorphism of complexes over $\mathfrak{Coh}_\alpha \times \mathfrak{Coh}_{\delta }$

\begin{align*} \begin{split} (\operatorname {RHom}\nolimits _S(\mathcal{E}_\delta , \mathcal{E}_{\alpha })[1])^\vee & = \operatorname {RHom}\nolimits _S(\mathcal{E}_\alpha , K_{S}\otimes \mathcal{E}_\delta )[1]\\ & =Rp_{12*}(\mathcal{E}_{\alpha }^\vee \otimes K_{S}\otimes \mathcal{O}_{\Delta _{S}} \otimes \rho )[1]\\ & =\mathcal{E}_{\alpha }^\vee \otimes K_S\otimes \rho [1] \end{split} \end{align*}

where $p_{12}:\mathfrak{Coh}_\alpha \times \mathfrak{Coh}_{\delta } \times S \to \mathfrak{Coh}_\alpha \times \mathfrak{Coh}_{\delta }$ is the projection. In particular, the complex $\mathcal{F}_\alpha$ is the restriction to $\mathfrak{Coh}_\alpha \times S$ of $(\operatorname {RHom}\nolimits _S(\mathcal{E}_\delta , \mathcal{E}_{\alpha })[1])^\vee$ . Let

\begin{align*}\tau : \mathbb{P}(\mathcal{E}_{\gamma }) \to \mathbb{P}(\mathcal{E}_{\gamma }) \times {S} ,\quad \overline {\tau } : \mathbb{P}(\mathcal{F}_{\alpha }) \to \mathbb{P}(\mathcal{F}_{\alpha }) \times {S}\end{align*}

be the diagonal morphisms, i.e., the morphisms making the following diagrams commutative:

(3.4)

Here, $pr_{{S}}$ is the projection to $S$ . We consider the following partial truncations

\begin{align*}\widetilde {\mathcal{C}oh}_{\delta ;\alpha }= \widetilde {\mathfrak{Coh}}_{\delta ;\alpha } \underset {\mathfrak{Coh}_\delta }{\times } \mathcal{C}oh_\delta , \quad \widetilde {\mathcal{C}oh}^{{\geqslant } 1}_{\delta ;\alpha }= \widetilde {\mathfrak{Coh}}^{{\geqslant } 1}_{\delta ;\alpha } \underset {\mathfrak{Coh}_\delta }{\times } \mathcal{C}oh_\delta .\end{align*}

The motivation to consider these derived stacks will become clear in Section 3.4. The notation is in conflict with (2.7), but its usage will be clear from the context. We have the following important result.

Recall that the classical truncation map $\mathcal{C}oh_\delta \to \mathfrak{Coh}_\delta$ induces an isomorphism in Borel-Moore homology (and likewise in relative Borel-Moore homology). To carry out the computation of length one Hecke operators, we may and will therefore use the partially truncated induction diagrams

\begin{gather*} \mathcal{C}oh_{\delta } \times \mathfrak{Coh}_\alpha ^{{\geqslant } 1} \xleftarrow {q_{\delta ,\alpha }} \widetilde {\mathcal{C}oh}_{\delta ;\alpha }^{{\geqslant } 1} \xrightarrow {p_{\delta ,\alpha }} \mathfrak{Coh}_{\gamma }^{{\geqslant } 1},\\ \mathfrak{Coh}_{\gamma }^{{\geqslant } 1} \times \mathcal{C}oh_{\delta } \xleftarrow {\overline {q}_{\gamma ,\delta }}\widetilde {\mathcal{C}oh}_{\delta ;\alpha }^{{\geqslant } 1} \xrightarrow {\overline {p}_{\gamma ,\delta }} \mathfrak{Coh}_{\alpha }^{{\geqslant } 1}, \end{gather*}

obtained by base change from (3.3).

Proof. We already know that $p$ is proper. Since both $\mathcal{E}_\gamma$ and $\mathcal{F}_\alpha$ have perfect amplitude in $[-1,0]$ over $\mathfrak{Coh}^{{\geqslant } 1} \times {\overline {S}}$ , by [Reference JiangJia25, Lemma 5.4] the maps $p'$ and $\overline {p}'$ are proper and quasi-smooth. Hence $\overline {p}$ is proper as well when $S$ is proper. Since $\mathfrak{Coh}_\alpha$ is an open substack of $\mathfrak{Coh}_{\alpha }({\overline {S}})$ , we may deduce the case of an arbitrary $S$ by base change. Indeed, note that there is a cartesian diagram

since for any extension $0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O}_x\to 0$ with $\mathcal{F}, \mathcal{E}$ of dimension ${\geqslant } 1$ , we have ${\rm supp}(\mathcal{E})={\rm supp}(\mathcal{F})$ . Next, we claim that the maps $q$ and $\overline {q}$ are the restrictions to suitable open substacks of the projections

\begin{align*}\mathbb{V}(\operatorname {RHom}\nolimits _S(\mathcal{E}_\delta ,\mathcal{E}_\alpha )[1]) \to \mathcal{C}oh_\delta \times \mathfrak{Coh}_\alpha ,\quad \mathbb{V}(\operatorname {RHom}\nolimits _S(\mathcal{E}_\gamma ,\mathcal{E}_\delta )) \to \mathfrak{Coh}_\gamma \times \mathcal{C}oh_\delta .\end{align*}

Indeed, the condition for a sheaf to be supported on $S \subset \overline {S}$ is open, as is the condition for a morphism $\mathcal{E}_\gamma \to \mathcal{E}_\delta$ to be surjective. As a consequence, $q,\overline {q}$ are both quasi-smooth.

We will prove a more general version of Corollary 3.3 in Section 7.1. Thanks to Corollary 3.3, the truncated induction diagrams yield two types of Hecke operators

\begin{gather*} \begin{split} T_+ =p_!\circ q^*: H_*(\mathcal{C}oh_\delta ,\mathbb{Q}) \otimes H_*(\mathcal{C}oh_{\alpha }^{{\geqslant } 1},\mathbb{Q}) \to H_*(\mathcal{C}oh_\gamma ^{{\geqslant } 1},\mathbb{Q}),\\ T_- =\overline {p}_!\circ \overline {q}^*: H_*(\mathcal{C}oh_\delta ,\mathbb{Q}) \otimes H_*(\mathcal{C}oh_{\gamma }^{{\geqslant } 1},\mathbb{Q}) \to H_*(\mathcal{C}oh_\alpha ^{{\geqslant } 1},\mathbb{Q}). \end{split} \end{gather*}

Considering $S$ -hyperbolic homology, we also define

(3.5) \begin{align} \begin{split} T^c_+ =r\circ p'_!\circ q^*: H^c_*(\mathcal{C}oh_\delta ,\mathbb{Q}) \otimes H_* (\mathcal{C}oh_{\alpha }^{{\geqslant } 1},\mathbb{Q}) \to H_*(\mathcal{C}oh_\gamma ^{{\geqslant } 1},\mathbb{Q}),\\ T^c_- =r\circ \overline {p}'_!\circ \overline {q}^*: H^c_*(\mathcal{C}oh_\delta ,\mathbb{Q}) \otimes H_*(\mathcal{C}oh_{\gamma }^{{\geqslant } 1},\mathbb{Q}) \to H_*(\mathcal{C}oh_\alpha ^{{\geqslant } 1},\mathbb{Q}), \end{split} \end{align}

where $r: H_*^c(S,\mathbb{Q}) \to \mathbb{Q}$ is the canonical degree 0 map. We will next use Proposition 3.2 to compute their action on tautological classes over suitable open substacks of $\mathfrak{Coh}^{{\geqslant } 1}$ .

3.4 Atlases for length one correspondences

We assume until Section 3.7 that $S={\overline {S}}$ is a projective surface. Let us fix a finite type open derived substack $\mathfrak{U} \subset \mathfrak{Coh}^{{\geqslant } 1}_{\gamma }$ and a locally free resolution

(3.6) \begin{align} 0\to \mathcal{E}_{-1} \to \mathcal{E}_0 \to \mathcal{E}_{\gamma }|_{\mathfrak{U}\times {\overline {S}}} \to 0. \end{align}

We keep the notations from the previous section. Let us explicitly describe the projectivization $\mathbb{P}(\mathcal{E}_{\gamma })$ in terms of the complex $\mathcal{E}_{-1} \to \mathcal{E}_0$ . Let $\mathbb{P}(\mathcal{E}_0)$ be the total space of the projective bundle associated to $\mathcal{E}_0$ and $\pi : \mathbb{P}(\mathcal{E}_0) \to \mathfrak{U} \times \overline {S}$ be the projection. The $\mathbb{C}$ -points of $\mathbb{P}(\mathcal{E}_0)$ parametrize triples $(\mathcal{W},y,\lambda )$ where $\mathcal{W} \in \mathfrak{U}$ is a coherent sheaf on $\overline {S}$ of dimension ${\geqslant } 1$ , $y \in \overline {S}$ and $\lambda : \mathcal{E}_0|_{(\mathcal{W},y)} \to \mathbb{C}$ is a nontrivial linear form, defined up to multiplication by a scalar. The morphism $\pi ^*(\mathcal{E}_0^\vee ) \to \pi ^*(\mathcal{E}_{-1}^\vee )$ yields a map $\mathcal{O}_{\mathbb{P}(\mathcal{E}_0)}(-1) \to \pi ^*(\mathcal{E}_{-1}^\vee )$ . Recall that, if $V$ is a finite-dimensional vector space, then $\mathbb{P}(V)$ parametrizes hyperplanes of $V$ , see Section 1.4. Unless specifically mentioned, all fiber products and tensor products are derived. This map can be viewed as a section

(3.7) \begin{align} s \in H^0(\mathbb{P}(\mathcal{E}_0),\pi ^*(\mathcal{E}_{-1}^\vee )(1)). \end{align}

The zero locus $Z(s)$ of $s$ parametrizes the triples $(\mathcal{W},y,\lambda ) \in \mathbb{P}(\mathcal{E}_0)$ for which the map $\lambda$ descends to $\mathcal{W}|_y$ . By Proposition 3.2, we have

\begin{align*}\widetilde {\mathcal{C}oh}_{\delta ;\alpha } \underset {\mathfrak{Coh}_{\gamma }}{\times } \mathfrak{U}=\mathbb{P}(\mathcal{E}_{\gamma }|_{\mathfrak{U}\times \overline S})\simeq \mathbb{P}(\mathcal{E}_0) \underset {\mathbb{V}(\pi ^*(\mathcal{E}_{-1}^\vee )(1))}{\times }\mathbb{P}(\mathcal{E}_0).\end{align*}

The derived fiber product is taken with respect to the sections $s$ and 0 of $\pi ^*(\mathcal{E}_{-1}^\vee )(1)$ . Let $\mathbb{P}(\mathcal{E}_0^{cl})$ be the classical projective bundle of $\mathcal{E}^{cl}_0$ over $\mathfrak{U}^{cl}\times {\overline {S}}$ . By [Reference JiangJia25, Proposition 4.21], the classical truncation of the derived stack $\mathbb{P}(\mathcal{E}_{\gamma }|_{\mathfrak{U}\times \overline S})$ is isomorphic to the zero locus in $\mathbb{P}(\mathcal{E}_0^{cl})$ of the section $s^{cl}$ , i.e., we have

\begin{align*}\mathbb{P}(\mathcal{E}_{\gamma }|_{\mathfrak{U}\times \overline S})^{cl} = Z(s^{cl}).\end{align*}

On the other hand, over the partial truncation $\mathcal{C}oh_{\delta } \times \mathfrak{Coh}_{\alpha }^{{\geqslant } 1}$ , the complex $\operatorname {RHom}\nolimits (\mathcal{E}_\delta , \mathcal{E}_{\alpha })[1]$ has perfect amplitude in $[0,1]$ . We fix a finite type open derived substack $\mathfrak{U}'$ of $\mathfrak{Coh}_{\alpha }^{{\geqslant } 1}$ with a presentation of the complex $\operatorname {RHom}\nolimits (\mathcal{E}_\delta , \mathcal{E}_{\alpha })[1]\big |_{\mathcal{C}oh_{\delta } \times \mathfrak{U}'}$

(3.8) \begin{align} 0\to \mathcal{V}_0 \to \mathcal{V}_1\to 0 \end{align}

Let $t:\mathbb{V}(\mathcal{V}_0) \to \mathcal{C}oh_{\delta } \times \mathfrak{U}'$ be the projection. The map $\mathcal{V}_0 \to \mathcal{V}_1$ yields a section

(3.9) \begin{align} s' \in H^0(\mathbb{V}(\mathcal{V}_0), t^*(\mathcal{V}_1)). \end{align}

By Section 2.3 there is an isomorphism

\begin{align*}\widetilde {\mathcal{C}oh}_{\delta ;\alpha } \underset {\mathfrak{Coh}_{\alpha }}{\times }\mathfrak{U}'= \mathbb{V}(\mathcal{V}_0) \underset {\mathbb{V}(t^*(\mathcal{V}_1))}{\times }\mathbb{V}(\mathcal{V}_0)\end{align*}

where the derived fiber product is taken with respect to the sections $s'$ and 0. By [Reference JiangJia25, Proposition 4.10], the classical truncation is isomorphic to the zero locus $Z((s')^{cl}) \subset \mathbb{V}(\mathcal{V}^{cl}_0)$ . We thus get the following isomorphisms of derived and classical stacks over any open set on which both presentations (3.6) and (3.8) exist:

(3.10) \begin{align} \begin{split} \mathbb{V}(\mathcal{V}_0) \underset {\mathbb{V}(t^*(\mathcal{V}_1))}{\times }\mathbb{V}(\mathcal{V}_0) & \simeq \widetilde {\mathfrak{Coh}}_{\delta ;\alpha } \simeq \mathbb{P}(\mathcal{E}_0) \underset {\mathbb{V}(\pi ^*(\mathcal{E}_{-1}^\vee )(1))}{\times }\mathbb{P}(\mathcal{E}_0),\\[5pt] \mathbb{V}(\mathcal{V}^{cl}_0) & \supset Z((s')^{cl}) = Z(s^{cl}) \subset \mathbb{P}(\mathcal{E}^{cl}_0). \end{split} \end{align}

The case of $\mathbb{P}(\mathcal{F}_\alpha )$ is similar, let us briefly sketch it. Observe that

\begin{align*}\operatorname {RHom}\nolimits _S(\mathcal{E}_\gamma ,\mathcal{E}_\delta )=Rp_{12*}(\mathcal{E}_\gamma ^\vee \otimes \mathcal{E}_\delta )=Rp_{12*}(\mathcal{E}_\gamma ^\vee \otimes \mathcal{O}_{\Delta _{{\overline {S}}}} \otimes \rho )=\mathcal{E}_\gamma ^\vee \otimes \rho .\end{align*}

Hence there is a factorization

where $j$ is an open embedding. Let us restrict everything to an open substack of finite type $\mathfrak{U}\subset \mathfrak{Coh}_\gamma ^{{\geqslant } 1}$ . Consider locally free resolution (3.6), and let $\overline {\pi }: \mathbb{V}(\mathcal{E}^\vee _\gamma \otimes \rho ) \to \mathfrak{U} \times \mathcal{C}oh_\delta$ be the projection. We have an obvious section

(3.11) \begin{align} \overline {s} \in H^0(\mathbb{V}(\mathcal{E}_0^\vee \otimes \rho ),\overline {\pi }^*(\mathcal{E}^\vee _{-1}\otimes \rho )). \end{align}

The zero locus of $\overline {s}^{cl}$ in $\mathbb{V}(\mathcal{E}_0^\vee \otimes \rho ^{cl})$ is

\begin{align*}\mathbb{V}(\mathcal{E}^{\vee }_\gamma \otimes \rho |_{\mathfrak{U} \times \mathcal{C}oh_\delta })^{cl} =Z(\overline {s}^{cl}) \end{align*}

Likewise, fix an open substack of finite type $\mathfrak{U}' \subset \mathfrak{Coh}_\alpha ^{{\geqslant } 1}$ as in (3.8), so that $\mathcal{F}_{\alpha }|_{ \mathfrak{U}'\times {\overline {S}}}$ has a presentation

\begin{align*} 0\to \mathcal{V}_1^\vee \to \mathcal{V}_0^\vee \to 0. \end{align*}

We have an obvious section

(3.12) \begin{align} \overline {s}' \in H^0(\mathbb{P}(\mathcal{V}_0^\vee ),\overline {p}^*(\mathcal{V}_1(1))), \end{align}

and the zero locus of $(\overline {s}')^{cl}$ in $\mathbb{P}(\mathcal{V}_0^{\vee ,cl})$ is $\mathbb{P}(\mathcal{F}_{\alpha }|_{\mathfrak{U}\times \overline S})^{cl} =Z((\overline {s}')^{cl})$ . Therefore, over any open set over which both presentations (3.6) and (3.8) exist, we have the following isomorphisms of derived and classical stacks

(3.13) \begin{align} \begin{split} \mathbb{P}(\mathcal{V}_0^\vee ) \underset {\mathbb{V}(\overline {p}^*(\mathcal{V}_1\otimes \rho ))}{\times }\mathbb{P}(\mathcal{V}_0^\vee ) & \simeq \widetilde {\mathcal{C}oh}_{\delta ;\alpha } \simeq \mathbb{V}(\mathcal{E}_0^\vee \otimes \rho ) \underset {\mathbb{V}(\overline {\pi }^*(\mathcal{E}_{-1}^\vee \otimes \rho ))}{\times }\mathbb{V}(\mathcal{E}_0^\vee \otimes \rho ),\\ \mathbb{P}(\mathcal{V}_0^{\vee ,cl}) & \supset Z((\overline {s}')^{cl}) \simeq Z(\overline {s}^{cl}) \subset \mathbb{V}(\mathcal{E}_0^{\vee ,cl}\otimes \rho ). \end{split} \end{align}

For the future use, note that the image of the map $j^{cl}$ is the complement of the zero section in $\mathbb{V}(\mathcal{E}_0^{\vee , cl}\otimes \rho )$ . Therefore, the section $\overline {s}^{cl}$ is regular over this complement if and only if the section $s^{cl}$ is regular. Similarly, the section $(\overline {s}')^{cl}$ is regular if and only if the section $(s')^{cl}$ is regular.

3.5 Computation of Hecke operators on fundamental classes

We are now in position to compute the action of Hecke operators on the fundamental classes $[\mathcal{C}oh^{{\geqslant } 1}_\alpha ], [\mathcal{C}oh^{{\geqslant } 1}_\gamma ]$ and on the virtual fundamental classes $[\mathfrak{Coh}^{{\geqslant } 1}_\alpha ], [\mathfrak{Coh}_\gamma ^{{\geqslant } 1}]$ . We keep the notation of the previous sections. Fix finite type open substacks $\mathfrak{U} \subset \mathfrak{Coh}^{{\geqslant } 1}_{\gamma }$ and $\mathfrak{U}' \subset \mathfrak{Coh}_{\alpha }^{{\geqslant } 1}$ such that

(3.14) \begin{align} q^{-1}(\mathfrak{Coh}_{\delta }({\overline {S}}) \times \mathfrak{U}') \supseteq p^{-1}(\mathfrak{U}). \end{align}

Recall that $\gamma =\alpha +\delta$ . We will carry out the computation of the action on the non-virtual fundamental classes first, under the following assumption

(3.15) \begin{align} \boxed{\textrm {the sections}\,\, s^{cl}\,\, \textrm {and}\,\, (s')^{cl}\,\, \textrm {are regular.}} \end{align}

This condition implies that the sections $\overline {s}^{cl}$ and $(\overline {s}')^{cl}$ in (3.11), (3.12) are regular as well. This means that the maps $q_{\delta ,\alpha }^{cl}$ and $p_{\delta ,\alpha }^{cl}$ in (3.1) are of the expected dimension over each irreducible component of $\mathfrak{U}^{cl}$ and $(\mathfrak{U}')^{cl}$ respectively. Recall from Section 2.2 the isomorphisms

\begin{align*}\mathcal{C}oh_\delta = {\overline {S}} \times B\mathbb{G}_m, \quad H^*(\mathcal{C}oh_\delta ,\mathbb{Q})=H^*({\overline {S}},\mathbb{Q})\otimes \mathbb{Q}[u],\end{align*}

where $u=c_1(\rho )$ is as in (2.3). Let

\begin{align*}[\Delta _{{\overline {S}}}] \in H^*(\mathcal{C}oh_\delta ,\mathbb{Q})\otimes H^*(\overline {S},\mathbb{Q})\end{align*}

be the fundamental class of the diagonal. Let $h_n$ be the complete symmetric function of degree $n$ . Recall the notation $h_n(\mathcal{E}_\gamma )$ in (1.6).

Before giving the proof, it is worth spelling out the meaning of formula (3.16). On the l.h.s., we view $\big (u^l[\Delta _{\overline S}] \cap [\mathcal{C}oh_\delta ]\big )\otimes [\mathcal{C}oh_{\alpha }^{{\geqslant } 1}]$ as an element of the triple tensor product $H_*(\mathcal{C}oh_\delta ,\mathbb{Q}) \otimes H^*(\overline {S},\mathbb{Q}) \otimes H_*(\mathcal{C}oh_\alpha ^{{\geqslant } 1})$ , and we apply the operator $T_+$ to the first and third factors. On the r.h.s., we view $h_{l+1-r}(\mathcal{E}_\gamma )$ as an element of $H^*(\mathcal{C}oh_\gamma ,\mathbb{Q}) \otimes H^*(\overline {S},\mathbb{Q})$ and act by cap product on the left term in the tensor product.

Proof. Let $j$ be the open immersion $p^{-1}(\mathfrak{U}) \subset q^{-1}(\mathcal{C}oh_{\delta } \times \mathfrak{U}')$ . We will work with classical stacks, but we will omit the superscript $cl$ in the notation. Under the assumption (3.15), we have

\begin{align*}j^*q^!([\mathcal{C}oh_\delta ] \otimes [\mathcal{C}oh_\alpha ])=j^*([Z(s')])=[Z(s)].\end{align*}

We abbreviate $\mathcal{O}(1)=\mathcal{O}_{\mathbb{P}(\mathcal{E}_0)}(1)$ . By Proposition 3.2, we have the following isomorphism of coherent sheaves over $p^{-1}(\mathfrak{U}) \times \overline {S}$ :

\begin{align*}(j \times {\rm Id})^*(q \times {\rm Id})^*\mathcal{E}_\delta =\tau _*\mathcal{O}(1)\end{align*}

We deduce that

(3.18) \begin{align} (j \times {\rm Id})^*(q \times {\rm Id})^!(({\rm ch}(\mathcal{E}_\delta ) \cap [\mathcal{C}oh_\delta \times \overline {S}] )\otimes [\mathcal{C}oh_\alpha ]) & ={\rm ch}(\tau _*\mathcal{O}(1)) \cap [Z(s) \times \overline {S}] \end{align}

Now, we consider the commutative diagram

The maps $\iota ,$ $ \iota '$ are the obvious closed immersions, the map $\pi$ is the projection to $\mathfrak{U} \times \overline {S}$ , the map $p'$ is the projection to $\mathfrak{U}$ , and the maps $\tau ,$ $\tau '$ are defined as in (3.4). The bottom row of the diagram clearly composes to $p\times {\rm Id}$ .

Let $i: \mathfrak{U} \to \mathfrak{Coh}^{{\geqslant } 1}_\gamma$ be the open immersion. Applying $(p\times {\rm Id})_*$ to (3.18) and restricting to the open subset $\mathfrak{U} \times \overline {S}$ , we get

\begin{align*} \begin{split} (i \times {\rm Id})^* & T_+\big (({\rm ch}(\mathcal{E}_\delta ) \cap [\mathcal{C}oh_\delta \times \overline {S}]) \otimes [\mathcal{C}oh_\alpha ^{{\geqslant } 1}]\big )\\ & = (p' \times {\rm Id})_*(\iota \times {\rm Id})_*\big ({\rm ch}(\tau _*\mathcal{O}(1)) \cap ([Z(s) \times \overline {S}])\big ) \end{split} \end{align*}

We claim that

(3.19) \begin{align} \tau _*\iota ^*\mathcal{O}(1)=(\iota \times {\rm Id})^*\tau '_*\mathcal{O}(1). \end{align}

Indeed, since $s$ is regular, the maps $\iota \times {\rm Id}$ and $\tau '$ are Tor-independent, and so we may use the proper base change theorem. Applying Chern character to (3.19), we get

\begin{align*} \begin{split} (i \times {\rm Id})^* & T_+\big (({\rm ch}(\mathcal{E}_\delta ) \cap [\mathcal{C}oh_\delta \times \overline {S}]) \otimes [\mathcal{C}oh_\alpha ^{{\geqslant } 1}])\big )\\ & = (p' \times {\rm Id})_*\big ({\rm ch}(\tau '_*\mathcal{O}(1)) \cap (\iota \times {\rm Id})_*([Z(s) \times \overline {S}])\big )\\ & =(p' \times {\rm Id})_*\big ({\rm ch}(\tau '_*\mathcal{O}(1)) \cap {\rm eu}\left (\pi ^*\mathcal{E}_{-1}^\vee (1)\right ) \cap [\mathbb{P}(\mathcal{E}_0)]\big )\\ & =(p' \times {\rm Id})_*\big ({\rm eu}\left (\pi ^*\mathcal{E}_{-1}^\vee (1)\right )\cap \tau '_*{\rm ch}(\mathcal{O}(1)) \cap {\rm Td}_{{\overline {S}}}^{-1} \cap [\mathbb{P}(\mathcal{E}_0)]\big )\\ & ={\rm Td}_{{\overline {S}}}^{-1} \cap (p' \times {\rm Id})_*\tau '_*\big ({\rm eu}\left (\pi ^*\mathcal{E}_{-1}^\vee (1)\right ) \cap {\rm ch}(\mathcal{O}(1)) \cap [\mathbb{P}(\mathcal{E}_0)]\big )\\ & ={\rm Td}_{{\overline {S}}}^{-1} \cap \pi _*\big ({\rm eu}\left (\pi ^*\mathcal{E}_{-1}^\vee (1)\right ) \cap {\rm ch}(\mathcal{O}(1)) \cap [\mathbb{P}(\mathcal{E}_0)]\big ) \end{split} \end{align*}

where we have successively used the fact that $s$ is regular and the Grothendieck-Riemann-Roch formula for the proper morphism $\tau$ . The following formula is well-known.

Lemma 3.5. Let $X$ be a stack. Let $\mathcal{E}_{-1},$ $ \mathcal{E}_0$ be vector bundles over $X$ . Set $r=\operatorname {rk}\nolimits (\mathcal{E}_0)-\operatorname {rk}\nolimits (\mathcal{E}_{-1})$ . Let $\pi : \mathbb{P}(\mathcal{E}_0) \to X$ be the projection. The following formula holds in $H^*(X,\mathbb{Q})$ :

\begin{align*} \pi _*\big ({\rm ch}(\mathcal{O}(1)) \cup {\rm eu}(\pi ^*\mathcal{E}_{-1}^\vee (1))\big )= \sum _{n\in \mathbb{N}}\frac {1}{n!}\,h_{n-r+1}(\mathcal{E}_0 - \mathcal{E}_{-1}). \end{align*}

Using the above lemma, we deduce that

(3.20) \begin{align} i^* T_+\Big (\big ({\rm ch}(\mathcal{E}_\delta ) \cap [\mathcal{C}oh_\delta ]\big ) \otimes [\mathcal{C}oh_{\alpha }^{{\geqslant } 1}]\Big )={\rm Td}_{{\overline {S}}}^{-1} \cap \sum _n \frac {1}{n!}h_{n+1-r}(\mathcal{E}_{\gamma }) \cap [\mathfrak{U}]. \end{align}

The proof of (3.16) now follows by multiplying throughout by the Todd class ${\rm Td}_{{\overline {S}}}$ and equating the terms of fixed homological degrees. Finally, the formula (3.17) is obtained by taking the intersection pairing with $\mu$ .

A similar analysis can be made for negative Hecke correspondences. Once again, we fix finite type open substacks $\mathfrak{U} \subset \mathfrak{Coh}^{{\geqslant } 1}_{\gamma }$ and $\mathfrak{U}' \subset \mathfrak{Coh}_{\alpha }^{{\geqslant } 1}$ satisfying

(3.21) \begin{align} \overline {q}^{-1}(\mathfrak{Coh}_{\delta }({\overline {S}}) \times \mathfrak{U}) \supseteq \overline {p}^{-1}(\mathfrak{U}')\quad \Leftrightarrow \quad q^{-1}(\mathfrak{Coh}_{\delta }({\overline {S}}) \times \mathfrak{U}') \subseteq p^{-1}(\mathfrak{U}). \end{align}

Recall the shift operation $\tau _c : \Lambda ' \to \Lambda '[c]$ defined by (1.5), and the notations in (2.12). We consider the algebra automorphism $x \mapsto \widetilde {x}$ of $\Lambda ({\overline {S}})$ such that

\begin{align*}\widetilde {f}(\lambda ) = \int _S p(\tau _{c_1}f) \cup \lambda \in \Lambda ({\overline {S}}) ,\quad f\in \Lambda ,\quad \lambda \in H^*({\overline {S}},\mathbb{Q}).\end{align*}

For any $\gamma \in K_0^c({\overline {S}})_{\mathbb{Q}}$ , we have

\begin{align*}\operatorname {ev}\nolimits _{\gamma }(\widetilde {f}(\lambda ))=\int _S f(\mathcal{E}_\gamma \otimes K^\vee _{\overline {S}}) \cup \lambda \end{align*}

where $f(\mathcal{E}_\gamma \otimes K^\vee _{\overline {S}})$ is as in (1.6).

Remark 3.6. For each $n\geqslant 0$ we have

\begin{align*} \widetilde {e}_n(\mu ) = \sum _{i=0}^n \binom {{\bf r}-n+i}{i} e_{n-i}(\mu \cup c_1^i). \end{align*}

In the absence of an equivariant parameter we have $c_1^3=0$ , and so only the first three terms of the sum above do not vanish. In particular, when $S$ has trivial canonical bundle we have

\begin{align*}\widetilde {e}_n(\mu ) = e_n(\mu ).\end{align*}

In the situation of Proposition 3.7 below we have ${\bf r}=r$ .

In this paper, we will check the assumption (3.15) in two situations of interest: the Hilbert schemes of points $\mathfrak{Hilb}_n(\overline {S})$ in Section 8.3, and the stacks of Higgs bundles $\mathfrak{Higgs}_{r,d}$ over a smooth projective curve in Section 9.2.

Let us now turn our attention to the action of Hecke operators on virtual fundamental classes.

Proposition 3.8. For any $\alpha$ , we have

\begin{align*} T_+\Big (\big (\mu u^l \cap [\mathcal{C}oh_\delta ]\big ) \otimes [\mathfrak{Coh}_{\alpha }]\Big )\Big |_{\mathfrak{U}} & = h_{l+1-r}(\mu ) \bullet [\mathfrak{U}],\\[5pt] T_-\Big (\big (\mu u^l \cap [\mathcal{C}oh_\delta ]\big ) \otimes [\mathfrak{Coh}_{\gamma }]\Big )\Big |_{\mathfrak{U}'} & = (-1)^{l}\widetilde {e}_{l+1+r}(\mu )\bullet [\mathfrak{U}']. \end{align*}

3.6 Hecke operators on tautological classes

Recall that $S={\overline {S}}$ is proper and that $u=c_1(\rho )$ is as in (2.3). In particular, the map $\omega _\delta : \Lambda (S) \to H_*(\mathcal{C}oh_\delta ,\mathbb{Q})$ in (2.19) is surjective by Lemma 2.19. Hence Proposition 3.4 allows us to describe the action of the space ${\bf H}_0(S)[1,-]$ on the fundamental class $[\mathcal{C}oh_\alpha ]$ after restriction to suitable open substacks $i:\mathfrak{U}_\alpha \to \mathfrak{Coh}_\alpha$ . Using the $\Lambda ({S})$ -module algebra structure of ${\bf H}(S)$ , we will now deduce a formula for the action of ${\bf H}_0(S)[1,-]$ on the subspace

\begin{align*}\Lambda ({S}) \bullet [\mathfrak{U}_\alpha ]\subset H^*(\mathfrak{U}_\alpha ,\mathbb{Q})\end{align*}

spanned by the tautological classes by length one Hecke operators. To do so, observe that there is a unique linear map

\begin{align*}\mathcal{L}^+_r : H^*(\mathcal{C}oh_\delta ,\mathbb{Q}) \otimes \Lambda ({S}) \to \Lambda ({S}) ,\quad r\geqslant 0\end{align*}

such that

1. (a) for any $\lambda \in H^*({S},\mathbb{Q})$ and $l \in \mathbb{N}$ , we have

   \begin{align*}\mathcal{L}^+_r(\lambda u^l \otimes 1)=h_{l+1-r}(\lambda ),\end{align*}

2. (b) for any $x,y \in \Lambda ({S})$ and $\xi \in H^*(\mathcal{C}oh_\delta ,\mathbb{Q})$ , we have

   \begin{align*}x \cdot \mathcal{L}^+_r(\xi \otimes y)= \sum \mathcal{L}^+_r( (\operatorname {ev}\nolimits _{\delta }(x^{(1)}) \cup \xi )\otimes ( x^{(2)} \cdot y))\end{align*}

where $\Delta (x)=\sum x^{(1)} \otimes x^{(2)}$ . By (3.17) and Proposition 2.18, we deduce that for $r=\operatorname {rk}\nolimits (\alpha )$ we have

(3.24) \begin{align} i^*\big ((\xi \cap [\mathcal{C}oh_\delta ]) \star ( x \bullet [\mathcal{C}oh_\alpha ])\big )=i^*\big (\mathcal{L}^+_r(\xi \otimes x) \bullet [\mathcal{C}oh_\gamma ]\big ) \end{align}

We will abbreviate

\begin{align*}\mathcal{L}^+_r(\xi ) : \Lambda ({S}) \to \Lambda ({S}) ,\quad x\mapsto \mathcal{L}^+_r(\xi \otimes x).\end{align*}

Then (b) translates into the following relation

(3.25) \begin{align} [p_n(\lambda ), \mathcal{L}^+_r(\xi )]=\mathcal{L}^+_r(\operatorname {ev}\nolimits _\delta (p_n(\lambda )) \cup \xi ). \end{align}

A direct computation using (2.1) and (2.4) gives

(3.26) \begin{align} \operatorname {ev}\nolimits _\delta (p_n(\lambda ))=f_n(u)\cup \lambda \end{align}

where $u=c_1(\rho )$ is as in (2.3) and

\begin{align*}f_n(u)=\frac {u^n - (u-t_1)^n - (u-t_2)^n + (u-t_1-t_2)^n}{t_1t_2}.\end{align*}

We define an algebra homomorphism $R^+$ and a $\Lambda ({S})$ -linear map $Q^+$ such that

(3.27) \begin{align} \begin{split} R^+ & : \Lambda ({S}) \to \Lambda ({S}) \otimes H^*(\mathcal{C}oh_\delta ,\mathbb{Q}) ,\quad p_n(\lambda )\mapsto p_n(\lambda )\otimes 1 - 1 \otimes \operatorname {ev}\nolimits _\delta (p_n(\lambda )),\\ Q^+ & : \Lambda ({S}) \otimes H^*(\mathcal{C}oh_\delta , \mathbb{Q}) \to \Lambda ({S}) ,\quad x \otimes \lambda u^l\mapsto x\cdot h_{l}(\lambda ). \end{split} \end{align}

We will extend $Q^+$ to a linear map

\begin{align*}Q^+: \Lambda (S) \otimes H^*(S,\mathbb{Q})[u,u^{-1}] \to \Lambda (S)\end{align*}

such that $Q^+(x \otimes u^l)=0$ for $l \lt 0$ . Then, the following formula holds

(3.28) \begin{align} \mathcal{L}^+_r(\lambda u^l)=L_{l+1-r}^+(\lambda ) \end{align}

where the right hand side is given by

(3.29) \begin{align} L^+_n(\lambda ): \Lambda (S)\to \Lambda (S), \quad f \mapsto Q^+(\lambda u^n R^+(f)) ,\quad n\in \mathbb{Z}. \end{align}

Again, it will be convenient to formally extend this definition to any $r \in \mathbb{Z}$ .

Using Proposition 3.7, we write similar formulas for negative correspondences. The analog of the relation (b) is

\begin{align*}x \cdot \mathcal{L}^-_r(\xi \otimes y)= \sum \mathcal{L}^-_r\big ( (\operatorname {ev}\nolimits _{\delta }(\upsilon (x^{(1)})) \cup \xi )\otimes (x^{(2)} \cdot y)\big ) ,\quad x,y \in \Lambda ({S}),\end{align*}

where $\upsilon$ is the involution (2.11). This translates to the relation

\begin{align*} [\widetilde {p}_n(\lambda ),\mathcal{L}_r^-(\xi )] = - \mathcal{L}_r^-(\operatorname {ev}\nolimits _\delta (\widetilde {p}_n(\lambda ))\cup \xi ). \end{align*}

Note that

\begin{align*} \operatorname {ev}\nolimits _\delta (\widetilde {p}_n(\lambda )) & = \int _S p_n(\mathcal{E}_\delta \otimes K^\vee _{\overline {S}})\cup \lambda \\[5pt] & = f_n(u+c_1)\cup \lambda \\[5pt] & = (-1)^n f_n(-u)\cup \lambda . \end{align*}

Thus we obtain the following formula

\begin{align*} \mathcal{L}^-_r(\lambda u^l)=(-1)^{r+1}\,L^-_{l+1+r}(\lambda )\end{align*}

where

(3.30) \begin{align} L^-_n(\lambda ): \Lambda (S)\to \Lambda (S), \quad f \mapsto Q^-(\lambda u^n R^-(f)) ,\quad n\in \mathbb{Z}. \end{align}

and $R^-$ and $Q^-$ are the algebra homomorphism and the $\Lambda (S)$ -linear map such that

\begin{align*} R^- & : \Lambda (S) \to \Lambda (S) \otimes H^*(\mathcal{C}oh_\delta ,\mathbb{Q}) ,\quad \widetilde {p}_n(\lambda )\mapsto \widetilde {p}_n(\lambda ) \otimes 1 + (-1)^n \otimes f_n(-u)\cup \lambda ,\\[5pt] Q^- & : \Lambda ({S}) \otimes H^*(\mathcal{C}oh_\delta ) \to \Lambda ({S}) ,\quad x \otimes \lambda u^l\mapsto (-1)^lx\cdot \widetilde {e}_{l}(\lambda ). \end{align*}

Propositions 3.4 and 3.7 yield the following.

Lemma 3.10. Let $r=\operatorname {rk}\nolimits (\alpha )$ . We have

\begin{align*} T_\pm ((\xi \cap [\mathcal{C}oh_\delta ])\otimes (f\bullet [\mathfrak{U}_\alpha ]))= \mathcal{L}^\pm _r(\xi )(f)\bullet [\mathfrak{U}_\alpha ] ,\quad \xi \in H^*(\mathcal{C}oh_\delta ) ,\quad f\in \Lambda (S). \end{align*}

For future use, we record here the following easily deduced formulas, valid for any $n$ and $\lambda$ :

(3.31) \begin{align} R^+(\widetilde {p}_n(\lambda )) & =\widetilde {p}_n(\lambda )\otimes 1 - (-1)^n \otimes f_n(-u)\cup \lambda \nonumber\\[5pt] R^-(p_n(\lambda )) & =p_n(\lambda ) \otimes 1+1 \otimes f_n(u)\cup \lambda . \end{align}

Remark 3.11. The operators $\mathcal{L}_r^\pm$ above are example of vertex operators. We have

\begin{align*} \begin{split} \sum _{\substack {n {\geqslant } 0\\ \gamma }} & \mathcal{L}_r^+(u^n\gamma )\otimes \gamma ^{*}s^{-n+r-1}\\ & =\Bigg\{\exp \Bigg( \sum _{\substack {n\gt 0\\\gamma }} \frac {p_n}{n}(\gamma )\otimes \gamma ^{*}s^{-n}\Bigg)_{[s^{\lt r}]} \exp \Bigg(-\sum _{\substack {n {\geqslant } 0\\\gamma }}\frac {\partial }{\partial \kappa _{n}(\gamma )} \otimes \gamma s^n\Bigg)\Bigg\}_{[s^{\lt r}]} \end{split} \end{align*}

\begin{align*} \begin{split} \sum _{\substack {n {\geqslant } 0\\ \gamma }} & \mathcal{L}_r^-(u^n\gamma )\otimes \gamma ^{*}s^{-n-r-1}\\ & =(-1)^{r+1} \Bigg\{\exp \Bigg(- \sum _{\substack {n\gt 0\\ \gamma }} \frac {\tau _{c_1}p_n}{n}(\gamma )\otimes \gamma ^{*}s^{-n}\Bigg)_{[s^{\lt -r}]} \exp \Bigg(\sum _{\substack {n {\geqslant } 0\\ \gamma }}\frac {\partial }{\partial \kappa _{n}(\gamma )} \otimes \gamma s^n\Bigg)\Bigg\}_{[s^{\lt -r}]} \end{split} \end{align*}

where $\{\gamma \},$ $\{\gamma ^*\}$ are dual bases of $H^*(S,\mathbb{Q})$ and the elements $\{\kappa _n(\lambda )\}$ are related to the $\{p_n(\lambda )\}$ through the relation

(3.32) \begin{align} \sum _{\substack {n {\geqslant } 0\\ \gamma }} \frac {x^{n+2}}{n!}\kappa _n(\gamma )\otimes \gamma ^*=\Bigg({\rm Td}_S(x) \cdot \sum _{\substack {n\gt 0\\ \gamma }} \frac {x^n}{n!}p_n(\gamma ) \otimes \gamma ^*\Bigg)_{[x^{\gt 1}]}. \end{align}

3.7 Hecke operators on open surfaces

Let us return to the situation when the surface $S$ is cohomologically pure but not necessarily proper. Pick a smooth compactification $\iota : S \to {\overline {S}}$ , and fix open substacks $\mathfrak{U} \subset \mathfrak{Coh}^{{\geqslant } 1}_{\gamma }({\overline {S}})$ and $\mathfrak{U}' \subset \mathfrak{Coh}_{\alpha }^{{\geqslant } 1}({\overline {S}})$ such thatFootnote ²

(3.33) \begin{align} \begin{split} q^{-1}(\mathfrak{Coh}_{\delta }(S) \times \mathfrak{U}') \supseteq p^{-1}(\mathfrak{U}), \\ \overline {q}^{-1}(\mathfrak{Coh}_{\delta }(S) \times \mathfrak{U}) \supseteq \overline {p}^{-1}(\mathfrak{U}'). \end{split} \end{align}

These conditions imply that the correspondences (3.1) restrict to Hecke operators

\begin{align*} T_+: H_*(\mathcal{C}oh_\delta (S),\mathbb{Q})\otimes H_*(\mathfrak{U}',\mathbb{Q})\to H_*(\mathfrak{U},\mathbb{Q}),\\ T_-: H_*(\mathcal{C}oh_\delta (S),\mathbb{Q})\otimes H_*(\mathfrak{U},\mathbb{Q})\to H_*(\mathfrak{U}',\mathbb{Q}) \end{align*}

The following simple lemma, which follows from open base change, relates these operators to the analogous operators for $\overline {S}$ . Let $j$ denote both inclusions $\mathfrak{U} \to \mathfrak{Coh}^{{\geqslant } 1}_{\gamma }$ and $\mathfrak{U}' \to \mathfrak{Coh}_{\alpha }^{{\geqslant } 1}$ .

Lemma 3.12. For $x \in H_*(\mathcal{C}oh_\delta ({\overline {S}}),\mathbb{Q})$ and $c \in H_*(\mathfrak{U},\mathbb{Q})$ , $c' \in H_*(\mathfrak{U}',\mathbb{Q})$ we have

\begin{align*} j^*T_+(x \otimes c')=T_+(\iota ^*x \otimes j^*c') \in H_*(\mathfrak{U},\mathbb{Q}),\\ j^*T_-(x \otimes c)=T_-(\iota ^*x \otimes j^*c) \in H_*(\mathfrak{U}',\mathbb{Q}). \end{align*}

Next, let us consider the compactly supported Hecke correspondences. Assume that the open substacks $\mathfrak{U} \subset \mathfrak{Coh}^{{\geqslant } 1}_{\gamma }$ and $\mathfrak{U}' \subset \mathfrak{Coh}_{\alpha }^{{\geqslant } 1}$ satisfy

(3.34) \begin{align} \begin{split} q^{-1}(\mathfrak{Coh}_{\delta }(S) \times \mathfrak{U}') &\supseteq (p')^{-1}(\mathfrak{U}\times S), \\ \overline {q}^{-1}(\mathfrak{Coh}_{\delta }(S) \times \mathfrak{U}) &\supseteq (\overline {p}')^{-1}(\mathfrak{U}'\times S). \end{split} \end{align}

In this case we can define restrictions of Hecke operators (3.5):

\begin{align*} T_+^c: H^c_*(\mathcal{C}oh_\delta (S),\mathbb{Q})\otimes H_*(\mathfrak{U}',\mathbb{Q})\to H_*(\mathfrak{U},\mathbb{Q}),\\ T_-^c: H^c_*(\mathcal{C}oh_\delta (S),\mathbb{Q})\otimes H_*(\mathfrak{U},\mathbb{Q})\to H_*(\mathfrak{U}',\mathbb{Q}). \end{align*}

Proposition 3.13. For $x \in H_*^c(\mathcal{C}oh_\delta (S),\mathbb{Q})$ and $c \in H_*(\mathfrak{U},\mathbb{Q})$ , $c' \in H_*(\mathfrak{U}',\mathbb{Q})$ we have

\begin{align*} j^*T_+(\iota _!(x)\otimes c') = T^c_+(x\otimes j^*c'),\quad j^*T_-(\iota _!(x)\otimes c) = T^c_-(x\otimes j^*c). \end{align*}

Proof. The two cases being identical, we will prove the statement for $T^c_+$ . Consider the following diagram in which we omit the obvious indices

Apart from the rightmost column, the second row is obtained from the top row by base change $-\times _{{\overline {S}}}S$ . In addition, all of the vertical arrows are open embeddings and the middle square in the bottom is cartesian. By Proposition A.6 (a), we have $q^!(\iota _! \otimes {\rm Id})=\tilde {\iota }_!t^!$ . Hence, using Proposition A.6 (b), we get

\begin{align*}p''_*p'_*q^!(\iota _!(x) \otimes c')=p''_*p'_*\tilde {\iota }_!t^!(x \otimes c')=p''_*({\rm Id} \otimes \iota _!)s'_*t^!(x \otimes c').\end{align*}

Note further that

\begin{align*}p''_*({\rm Id} \otimes \iota _!)=r : H_*^c(S,\mathbb{Q}) \to \mathbb{Q},\end{align*}

see Example A.5. It follows that

\begin{align*}T_+(\iota _!(x) \otimes c')=rs'_*t^!(x \otimes c').\end{align*}

On the other hand, by base change and functoriality of Gysin pullbacks, we have

\begin{align*}j^*s'_*t^!=(p')^\circ _*\widetilde {j}^*t^!=(p')^\circ _*(q^\circ )^!({\rm Id} \otimes j^*).\end{align*}

Thus, we have

\begin{align*} rs'_*t^!(x \otimes c')=r(p')^\circ _*(q^\circ )^!(x \otimes j^*(c'))=T^c_+(x\otimes j^*(c')). \end{align*}

One particular instance when both restrictions of Hecke operators are well defined is when

\begin{align*}\mathfrak{U} = \mathcal{C}oh_\gamma ^{{\geqslant } 1} ,\quad \mathfrak{U}' = \mathcal{C}oh_\alpha ^{{\geqslant } 1}.\end{align*}

If we further restrict to an appropriate open substack of $\mathcal{C}oh^{{\geqslant } 1}$ where the regularity conditions (3.15) hold, Proposition 3.13 and Lemma 3.12 imply that the formulas of Section 3.6 apply verbatim to the restricted operators $T_\pm$ , $T_\pm ^c$ over $\mathfrak{U}'$ , $\mathfrak{U}$ . Note that proper support implies $r=0$ . Since the evaluation map $\operatorname {ev}\nolimits$ factors through $\Lambda (S)$ by Lemma 2.16, in this case we can interpret the operators $\mathcal{L}^\pm _0$ as acting on $\Lambda (S)$ .

4.1 Positive halves of deformed $W$ -algebras

In what follows, we consistently write $\lambda \mu$ for the cup product $\lambda \cup \mu$ , in order to simplify the notation.

Definition 4.1. Let $W^{\geqslant }(S)$ be the $\mathbb{N}\times \mathbb{Z}$ -graded associative algebra generated by

\begin{align*}\psi _n(\lambda ) ,\quad T_n(\lambda ) ,\quad n {\geqslant } 0 ,\quad \lambda \in H^*(S,\mathbb{Q})\end{align*}

and a central element ${\bf c}$ modulo the following relations for $a,b \in \mathbb{Q}$ , $n,m \in \mathbb{N}$ , $\lambda ,\mu \in H^*(S,\mathbb{Q})$

(a) \begin{align} \psi _m(a\lambda +b\mu ) & = a\psi _m(\lambda )+b\psi _m(\mu ), \end{align}

(b) \begin{align} T_n(a\lambda +b\mu ) & = aT_n(\lambda )+bT_n(\mu ), \end{align}

(c) \begin{align} [\psi _m(\lambda ), \psi _n(\mu )] & = 0, \end{align}

(d) \begin{align} [\psi _m(\lambda ), T_n(\mu )] & = m T_{m+n-1}(\lambda \mu ), \end{align}

(e) \begin{align} [T_m(\lambda \mu ), T_n(\nu )] & = [T_m(\lambda ), T_n(\mu \nu )], \end{align}

(f) \begin{align} [T_{m}(\lambda ), T_{n+3}(\mu )] -3 [T_{m+1}(\lambda ), T_{n+2}(\mu )] \, +\,& 3 [T_{m+2}(\lambda ), T_{n+1}(\mu )] - [T_{m+3}(\lambda ), T_{n}(\mu )]\nonumber \\ -\ [T_{m}(\lambda ),{\kern1pt} T_{n+1}(s_2 \mu )]{\kern1pt} + [& T_{m+1}(\lambda ), T_{n}(s_2 \mu )] + \{T_m, T_n\}(c_1\Delta _S\lambda \mu )=0, \end{align}

(g) \begin{align} \sum _{w\in S_3} w \cdot [T_{m_3}(\lambda _3), &\, [T_{m_2}(\lambda _2), T_{m_1+1}(\lambda _1)]]=0, \end{align}

(h) \begin{align} \psi _n(\lambda )=0{\rm \ if\ } & 2n-2+\deg (\lambda )\lt 0. \end{align}

Recall that $[-,-]$ and $\{-,-\}$ are as in (1.3). The expression $\{T_m, T_n\}(c_1\Delta _S\lambda \mu )$ above is the anti-super-commutator of $T_m$ and $T_n$ , whose arguments are taken from the symmetric 2-tensor

\begin{align*}c_1\Delta _S\lambda \mu \in H^*(S,\mathbb{Q})\otimes H^*(S,\mathbb{Q}).\end{align*}

The generators have the following degrees

(4.1) \begin{align} \deg (T_n(\lambda ))=(1,2n-2+\deg (\lambda )), \quad \deg (\psi _n(\lambda ))=(0, 2n-2+\deg (\lambda )) \end{align}

and $\deg ({\bf c})=(0,0)$ .

Let $W^{\geqslant }(S)[m,n]$ be the subspace spanned by all bidegree $(m,n)$ elements. We define $W^+(S)$ and $W^0(S)$ to be the graded subalgebras generated by

\begin{align*}\{T_n(\lambda )\,:\,n {\geqslant } 0\,,\,\lambda \in H^*(S,\mathbb{Q})\} ,\quad \{{\bf c}, \psi _n(\lambda )\,:\,n {\geqslant } 0\,,\,\lambda \in H^*(S,\mathbb{Q})\}.\end{align*}

Note that $W^0(S)$ is a supercommutative algebra generated by elements indexed by pairs $(n,\lambda )$ . It will be convenient to identifyFootnote ³ $\Lambda (S)$ and $W^0(S)$ as follows. Set

\begin{align*} \psi _\lambda (z) & =\sum _{n {\geqslant } 0} \psi _n(\lambda ) \frac {z^n}{n!},\\ \underline {\psi }(z) & =z^{-1}{\bf c} \otimes 1 +\sum _\lambda \psi _\lambda (z) \otimes \lambda ^* \in W^0(S) \otimes H^*(S,\mathbb{Q})[[z]]. \end{align*}

There is a unique graded algebra isomorphism

\begin{align*}i: W^0(S) \simeq \Lambda (S) ,\quad \underline {\psi }(z) \mapsto z^{-1}{\bf r} \otimes 1 +z^{-1}(\underline {{\rm ch}}(z)-{\bf r} \otimes 1)(1\otimes {\rm Td}_S(z)).\end{align*}

Through this identification, we may consider the elements

\begin{align*}p(\lambda ) \in W^0(S) ,\quad p\in \Lambda ' ,\quad \lambda \in H^*(S,\mathbb{Q}).\end{align*}

Definition 4.3. Assume that $c_1=0$ and $s_2=q^2$ for some $q \in H^2(S,\mathbb{Q})$ . We define the Lie algebra $\mathfrak{w}^{\geqslant }(S)$ to be spanned by $z^mD^n\lambda$ with $m,n\in \mathbb{N}$ and $\lambda \in H^*(S,\mathbb{Q})$ , and a central element ${\bf c}$ , with the Lie bracket given by

(4.3) \begin{align} [z^mD^n\lambda \,,\,z^aD^b\mu ] = \frac { (D+aq)^nD^b - D^n(D+mq)^b }{q}\,z^{m+a}\lambda \mu . \end{align}

Let $\mathfrak{w}^+(S)$ be the subalgebra spanned by $z^mD^n\lambda$ for all $m {\geqslant } 1,$ $ n {\geqslant } 0$ and $\lambda \in H^*(S,\mathbb{Q})$ .

Remark 4.4. The right hand side of (4.3) is

\begin{align*} \sum _{i=1}^n \binom {n}{i}n^iz^{m+a}D^{n+b-i}q^{i-1}\lambda \mu - \sum _{j=1}^b \binom {b}{j}m^jz^{m+a}D^{n+b-j}q^{j-1}\lambda \mu . \end{align*}

The $q$ -term of this sum only depends on $(m+a)$ and $(mb-na)$ in accordance with the last claim of Proposition 4.8 below.

When $c_1=s_2=0$ , we write $D_{m,n}(\lambda )=z^mD^n\lambda$ . Then, the Lie bracket degenerates as follows:

(4.4) \begin{align} [D_{m,n}(\lambda ), D_{m',n'}(\mu )] = (n m' - m n') D_{m+m',n+n'-1}(\lambda \mu ). \end{align}

We will denote this Lie algebra by $\mathfrak{w}^{\geqslant }_0(S)$ . It is well-defined for any surface $S$ .

Theorem 4.5. If $c_1=0$ and $s_2=q^2$ , then the assignment $T_n(\lambda ) \mapsto zD^n\lambda$ extends to isomorphisms

\begin{align*}\Phi :W^{\geqslant }(S) \to U(\mathfrak{w}^{\geqslant }(S)) ,\quad \Phi :W^+(S) \to U(\mathfrak{w}^+(S)).\end{align*}

We will prove Theorem 4.2 in Sections 4.7 and 4.5, and Theorem 4.5 in Section 4.8.

4.2 Structure of $W^{\geqslant } (S)$ in the non-deformed case

In this section we assume that $c_1=s_2=0$ . So, the relation (f) simplifies to

(4.5) \begin{align} [T_{m}(\lambda ), T_{n+3}(\mu )] & - 3 [T_{m+1}(\lambda ), T_{n+2}(\mu )]+3 [T_{m+2}(\lambda ), T_{n+1}(\mu )] - [T_{m+3}(\lambda ), T_{n}(\mu )]=0. \end{align}

Theorem 4.6. If $c_1=s_2=0$ , then the assignment

(4.6) \begin{align} \psi _n(\lambda )\mapsto D_{0,n}(\lambda ) ,\quad T_n(\lambda )\mapsto D_{1,n}(\lambda ) ,\quad {\bf c} \mapsto {\bf c} \end{align}

extends to an algebra isomorphism

\begin{align*}\Phi : W^{\geqslant }(S)\to U(\mathfrak{w}_0^{\geqslant }(S)).\end{align*}

Proof. We abbreviate $T_n=T_n(1)$ and $\psi _n=\psi _n(1)$ . The elements $D_{0,n}(\lambda )$ , $D_{1,n}(\lambda )$ satisfy the relations (a)–(g). Hence the assignment above yields an algebra homomorphism. Set

\begin{align*}\widetilde {D}_{m,n}(\lambda ) = \frac {n!}{(m+n)!} (-\operatorname {Ad}\nolimits _{T_0})^m \psi _{m+n}(\lambda ).\end{align*}

We claim that the following defines an inverse homomorphism $\Psi$ to the map $\Phi$

(4.7) \begin{align} \Psi : D_{m,n}(\lambda ) \mapsto \widetilde {D}_{m,n}(\lambda ) . \end{align}

To prove the theorem, we need to check that the elements $\widetilde {D}_{m,n}(\lambda )$ satisfy the defining relations (4.4). They are satisfied for $m=n=0$ . We begin by computing a few commutators of low rank. Since $T_n$ is even, from (e) we deduce that

\begin{align*}[T_n,T_n(\lambda )]=[T_n(\lambda ),T_n]=0\end{align*}

Unraveling the definition of $\widetilde {D}_{m,n}(\mu )$ we obtain

(4.8) \begin{align} [T_0(\lambda ),\widetilde {D}_{m,n}(\mu )] = -n\widetilde {D}_{m+1,n-1}(\lambda \mu ), \quad n{\geqslant } 1. \end{align}

In the same way, the relations (d), (c) imply that

\begin{align*}[\psi _1(\lambda ), T_0(\mu )] = T_0(\lambda \mu ) ,\quad [\psi _1(\lambda ),\psi _n(\mu )]=0.\end{align*}

So we get

(4.9) \begin{align} [\psi _1(\lambda ),\widetilde {D}_{m,n}(\mu )] = m\widetilde {D}_{m,n}(\lambda \mu ). \end{align}

Let us now consider the commutators $[T_m(\lambda ),T_n(\mu )]$ . By (e), it’s enough to assume that $\lambda=1$ . Using (4.5) first for $m \in \{n,n \pm 1\}$ and then successively for $m\gt n$ and $m\lt n$ we deduce that for any $n$ , the commutators $[T_m,T_{n-m}(\mu )]$ for $m=0, 1, \ldots , n$ are all proportional to one another, hence, say to $[T_n,T_0(\mu )]=n\widetilde {D}_{2,n-1}(\mu )$ . But since $\Phi (\widetilde {D}_{2,m}(\mu ))=D_{2,m}(\mu )$ , we deduce that

\begin{align*}[T_m(\lambda ),T_{n-m}(\mu )]=[T_m,T_{n-m}(\lambda \mu )]=(2m-n)\widetilde {D}_{2,n-1}(\lambda \mu )\end{align*}

as expected. Further taking commutators with $\psi _m(\mu ) = \widetilde {D}_{0,m}(\mu )$ , we get

(4.10) \begin{align} [\widetilde {D}_{2,n}(\lambda ),\widetilde {D}_{0,m}(\mu )]=-2m\widetilde {D}_{2,n+m-1}(\lambda \mu ). \end{align}

Next, we consider commutators with $\widetilde {D}_{2,0}$ . Relation (g) yields

\begin{align*}[\widetilde {D}_{2,0},T_0] = [[T_1,T_0],T_0]=0,\end{align*}

and more generally

\begin{align*}[\widetilde {D}_{2,0}(\lambda ),T_0(\mu )]=0.\end{align*}

Using (4.8), we deduce from (4.10) by induction on $m$ that

(4.11) \begin{align} [\widetilde {D}_{2,0}(\lambda ),\widetilde {D}_{m,n}(\mu )] & = -\frac {1}{n+1}{[T_0,[\widetilde {D}_{2,0}(\lambda ),\widetilde {D}_{m-1,n+1}(\mu )]]} \\ & = 2[T_0,\widetilde {D}_{m+1,n}(\lambda \mu )] \nonumber \\ & = -2n \widetilde {D}_{m+2,n-1}(\lambda \mu )\nonumber \end{align}

for any $n {\geqslant } 1$ . Finally, from relations (4.11), (4.8), (4.9) and the definition of $\widetilde {D}_{m,n}$ we deduce in turn the following equalities

(4.12) \begin{align} [T_1(\lambda ),\widetilde {D}_{m,n}(\mu )] & = (m-n)\widetilde {D}_{m+1,n}(\lambda \mu ), \end{align}

(4.13) \begin{align} [\psi _2(\lambda ),\widetilde {D}_{m,n}(\mu )] & = 2m \widetilde {D}_{m,n+1}(\lambda \mu ) \end{align}

for all $m,n{\geqslant } 0$ . Assume now that the relations (4.4) hold (with $\widetilde {D}$ in place of $D$ ) for a given pair $(m,n)$ and all pair $(m',n')$ with $n'{\geqslant } 1$ . Then, by applying $\operatorname {Ad}\nolimits _{T_1}$ or $\operatorname {Ad}\nolimits _{T_0}$ , resp. $\operatorname {Ad}\nolimits _{\psi _2}$ and using (4.12) or (4.8), resp. (4.13), we deduce that the same is true for the pairs $(m+1,n)$ , resp. $(m,n+1)$ . Starting from $m=1,n=0$ , we deduce that (4.4) holds for any pairs $(m,n)$ , $(m',n')$ with $m {\geqslant } 1$ and $n'{\geqslant } 1$ , and thus, by symmetry, whenever $m'{\geqslant } 1$ and $n{\geqslant } 1$ . The only remaining cases occur when $m=m'=0$ (for which (4.4) trivially holds) and when $n=n'=0$ , which we now deal with. We will prove by induction on $m+m'$ that

(4.14) \begin{align} [\widetilde {D}_{m,0}(\lambda ),\widetilde {D}_{m',0}(\mu )]=0. \end{align}

Fix $s$ and assume that (4.14) holds whenever $m+m'{\leqslant } s$ . Note that the above calculations show that this is indeed the case for $s=3$ . If $m,m'\gt 0$ satisfy $m+m'=s+1$ then by the induction hypothesis and what we’ve already established we have

\begin{align*} [T_0,\widetilde {D}_{m-1,1}(\lambda )] & =-\widetilde {D}_{m,0}(\lambda ), \\ [T_0,\widetilde {D}_{m'-1,1}(\mu )] & =-\widetilde {D}_{m',0}(\mu ),\\ [T_0,\widetilde {D}_{m,0}(\lambda )] & =0,\\ [T_0,\widetilde {D}_{m',0}(\mu )] & =0,\\ \frac {1}{m'}{[\widetilde {D}_{m-1,1}(\lambda ),\widetilde {D}_{m',0}(\mu )]} & =\widetilde {D}_{m+m',0}(\lambda \mu )\\ & =-\frac {1}{m}{[\widetilde {D}_{m,0}(\lambda ),\widetilde {D}_{m'-1,1}(\mu )]}. \end{align*}

Applying $\operatorname {Ad}\nolimits _{T_0}$ to this last equation we deduce $[\widetilde {D}_{m,0}(\lambda ),\widetilde {D}_{m',0}(\mu )]=0$ . We are done.

4.3 Order filtration

Let us return to the deformed case, with $S$ a projective surface. Let $F_n$ , $n\in \mathbb{Z}$ , be the smallest increasing filtration of $W^{\geqslant }(S)$ such that ${\bf c} \in F_0$ and

1. – $\psi _n(\lambda ), T_n(\lambda )\in F_n$ for all $n\in \mathbb{N}$ , $\lambda \in H^*(S,\mathbb{Q})$ ,

2. – $F_nF_{n'} \subset F_{n+n'}$ ,

3. – $[F_n,F_{n'}]\subset F_{n+n'-1}$ .