2.1 Brauer–Severi varieties, Azumaya algebras and gerbes

In this section, we recall the notions of Brauer–Severi variety, Azumaya algebra and gerbe, and gather various known facts about them. Let $X$ be a smooth projective variety over $\mathbb{C}$ .

We will recall below that the data of a Brauer–Severi variety over $X$ is equivalent to the data of an Azumaya algebra on $X$ . We first recall the following version of the Skolem–Noether theorem [Reference MilneMil80, Proposition IV.2.3].

The facts in the following lemma are all proved in [Reference YoshiokaYos06, §1.1].

Lemma 2.4. If $\pi : Y \to X$ is a Brauer–Severi variety with relative tangent sheaf $T_{Y/X}$ , then there is a unique (up to scaling) sheaf $G$ fitting in a non-trivial short exact sequence

(2) \begin{align} 0 \to \mathcal{O}_Y \to G \to T_{Y/X} \to 0. \end{align}

Furthermore, the sheaf $G$ satisfies the properties $R \pi _*(G^\vee ) = 0$ and the canonical map $\pi ^* \pi _* (\mathcal{E}{\textit{nd}}(G^\vee )) \to \mathcal{E}{\textit{nd}}(G^\vee )$ is an isomorphism, where $\mathcal{E}{\textit{nd}}(G^\vee ) := G^\vee \otimes G$ .

We now state the following classification result on Brauer–Severi varieties and Azumaya algebras.

We only provide a sketch of the proof.

There is also a theory of gerbes when $A$ is non-abelian, which we will not need, but see [Reference GiraudGir71], where the general theory is developed. We require the following facts from the theory of gerbes, which can all be found in [Reference GiraudGir71, Chapter IV]:

As a consequence of the proposition, and the fact that gerbes admit local sections, every $A$ -gerbe on $X$ is étale locally isomorphic to the stack of $A$ -torsors $[*/A] \times X$ . The converse is almost true: $[*/A]$ is a group stack, and a stack over $X$ that is étale locally isomorphic to $[*/A] \times X$ is an $A$ -gerbe on $X$ if the gluing maps are $[*/A]$ -equivariant.

From now on, we will only be concerned with $\mathbb{G}_m$ - and $\mu _r$ -gerbes.

By the previous examples, every Brauer–Severi variety $Y$ on $X$ of degree $r$ defines classes $\alpha (Y) \in H^2(X, \mathbb{G}_m)$ and $w(Y) \in H^2(X, \mu _r)$ . We refer to $\alpha (Y)$ as the Brauer class of $Y$ and to $w(Y)$ as the Stiefel–Whitney class of $Y$ . Similarly, every Azumaya algebra $\mathcal{A}$ on $X$ of degree $r$ determines classes $\alpha (\mathcal{A}) \in H^2(X, \mathbb{G}_m)$ and $w(\mathcal{A}) \in H^2(X, \mu _r)$ . The assignments $Y \mapsto w(Y)$ and $Y \mapsto \alpha (Y)$ fit in the framework of non-abelian cohomology [Reference GiraudGir71]. This theory defines groups $H^1(X, G)$ and $H^2(X, G)$ for non-abelian groups $G$ in the étale topology, and defines connecting homomorphisms between these groups. The definitions of the connecting homomorphisms induced by the sequences

(3) \begin{align} 1 \to \mu _r \to{{\rm SL}}_r \to{{\rm PGL}}_r \to 1 \qquad \ \mathrm{and}\ \qquad 1 \to \mathbb{G}_m \to{{\rm GL}}_r \to{{\rm PGL}}_r \to 1 \end{align}

are precisely $w$ and $\alpha$ . Furthermore, these maps are compatible in the sense that the following diagram commutes:

(4)

where $o$ is induced by the inclusion $\mu _r \leq \mathbb{G}_m$ . This can be seen by either using that there is a morphism between the short exact sequences from (3), or, alternatively, by explicitly comparing the gerbes. Note that the vertical arrow $o$ in the diagram is part of the long exact sequences induced by the Kummer sequence

(5) \begin{align} \cdots \to H^1(X,\mathbb{G}_m) \to H^2(X,\mu _r) \to H^2(X,\mathbb{G}_m) \stackrel{(\cdot )^r}{\to } H^2(X,\mathbb{G}_m) \to \cdots. \end{align}

Since $H^1(X,\mathbb{G}_m) \cong {\rm Pic}(X)$ , we refer to the first map as $c_1 : H^1(X,\mathbb{G}_m) \to H^2(X,\mu _r)$ . In the trivial case we get [Reference YoshiokaYos06, Lemma 1.2], [Reference Huybrechts and SchröerHS03]

(6) \begin{align} w(\mathbb{P}(E^\vee )) = [c_1(E) \mod r] \in H^2(X, \mu _r). \end{align}

By the comparison theorem [Reference MilneMil80], the étale cohomology group $H^2(X,\mu _r)$ equals the corresponding singular cohomology group with respect to the complex analytic topology on $X$ , which we sometimes denote by $H^2(X,\mathbb{Z} / r\mathbb{Z})$ . The above factorization (4) also reveals that the Brauer class of a degree $r$ Azumaya algebra on $X$ is an $r$ -torsion element of $H^2(X,\mathbb{G}_m)$ .

In the next section, we assume $H_1(X,\mathbb{Z})_{{\rm tor}} = 0$ (in the complex analytic topology). Then, by Lemma 3.1, we have

\begin{align*} H^2(X,\mu _r) \cong H^2(X,\mathbb {Z}) / rH^2(X,\mathbb {Z}). \end{align*}

Using the identification $H^1(X,\mathbb{G}_m) \cong {\rm Pic}(X)$ , the map $H^1(X,\mathbb{G}_m) \to H^2(X,\mu _r)$ factors via the usual first Chern class map

where the diagonal arrow is the quotient map and the vertical arrow is surjective by the Lefschetz theorem on $(1,1)$ -classes. By the Kummer sequence (5), the $\mu _r$ -gerbes which are trivial as $\mathbb{G}_m$ -gerbes are precisely the elements of

\begin{align*} (H^2(X,\mathbb {Z}) \cap H^{1,1}(X)) / r H^2(X,\mathbb {Z}). \end{align*}

The cohomology class $\alpha (Y) \in H^2(X, \mathbb{G}_m)$ determines whether $Y$ is trivial (in the sense of Example 2.6), because $\alpha (Y)=0$ if and only if the gerbe from Example 2.13 has a global section (Proposition 2.10), but this precisely happens when a trivialising locally free sheaf exists on $X$ . This statement can also be seen from the previous paragraph combined with [Reference YoshiokaYos06, Lemma 1.4].Footnote ¹⁰

It is not possible to recover a Brauer–Severi variety from its class in $H^2(X, \mathbb{G}_m)$ ; indeed, all Brauer–Severi varieties of the form $\mathbb{P}(E^\vee )$ for some locally free sheaf $E$ have trivial class. Thus, taking the associated class is not ‘injective’. Instead, we have the following ’surjectivity‘ results.

Theorem 2.15 holds far more generally. We only need $X$ to admit an ample line bundle, for example, if $X$ is quasi-projective over an affine scheme (see [Reference de JongdJo]). On the contrary, Theorem 2.16 is special in the sense that it does not hold for smooth varieties of arbitrary dimension [Reference Antieau and WilliamsAW14]. It is a consequence of the period-index theorem proved by de Jong [Reference de JongdJo04]. See [Reference LieblichLie08, Corollary 4.2.2.4] on how to deduce this specific form of the theorem.

2.2 Twisted sheaves

We review the theory of twisted sheaves from different viewpoints. Recall the sheaf $G$ from Lemma 2.4.

The category of $Y$ -sheaves only depends on $\alpha (Y) \in H^2(X, \mathbb{G}_m)$ . We review the theory of coherent sheaves on gerbes and explain how to recover ${{\rm Coh}}(X,\mathcal{A})$ , ${{\rm Coh}}(X,Y)$ . The following result can be found in work of Lieblich [Reference LieblichLie07, Propositions 2.1.1.13, 2.2.1.6].

Proposition 2.19. (Lieblich). Let $\pi : \mathcal{G} \to X$ be a $\mathbb{G}_m$ -gerbe. Every quasi-coherent sheaf $F$ on $\mathcal{G}$ admits a canonical $\mathbb{G}_m$ -action. Hence, it admits a functorial weight decomposition

\begin{align*} F = \bigoplus _{n \in \mathbb {Z}} F_n. \end{align*}

In fact, the entire category of quasi-coherent sheaves admits a weight decomposition

\begin{align*} \mathrm {QCoh}(\mathcal {G}) = \bigoplus _{n \in \mathbb {Z}} \mathrm {QCoh}(\mathcal {G})_n. \end{align*}

Furthermore, the pull-back $\pi ^* : {\rm QCoh}(X) \to {\rm QCoh}(\mathcal{G})$ induces an equivalence between ${\rm QCoh}(X)$ and ${\rm QCoh}(\mathcal{G})_0$ , these are the sheaves on which the action is trivial. For a $\mu _r$ -gerbe the same result holds, except that the grading is now over $\mathbb{Z}/r\mathbb{Z}$ .

Tensor product and $\mathcal{H}{\textit{om}}$ respects this decomposition, in the sense that if $F \in {\rm QCoh}(\mathcal{G})_n$ and $F^{\prime} \in {\rm QCoh}(\mathcal{G})_m$ , then $F \otimes F^{\prime} \in {\rm QCoh}(\mathcal{G})_{n+m}$ and $\mathcal{H}{\textit{om}}(F, F^{\prime}) \in {\rm QCoh}(\mathcal{G})_{-n+m}$ . In particular, if $F$ is of pure weight, then $\mathcal{H}{\textit{om}}(F, F)$ is of weight zero and thus descends to $X$ .

Let $\mathcal{A}$ be an Azumaya algebra on $X$ mapping to a $\mathbb{G}_m$ -gerbe $\pi : \mathcal{G} \to X$ . Then there exists a rank $r$ locally free sheaf $E$ on $\mathcal{G}$ such that $\mathcal{E}{\textit{nd}}(E^\vee ) \cong \mathcal{A}|_{\mathcal{G}}$ . One can verify that $E$ is in fact a twisted sheaf [Reference LieblichLie07, Corollary 2.2.2.2]. If $F$ is any $\mathcal{G}$ -twisted sheaf, then $\mathcal{H}{\textit{om}}(E, F)$ is again untwisted, i.e. it is (uniquely) in the image of $\pi ^*$ by Proposition 2.19. Note that $\mathcal{H}{\textit{om}}(E, F)$ is a right $\mathcal{E}{\textit{nd}}(E)$ -module and therefore a left $\mathcal{E}{\textit{nd}}(E)^{{\rm op}} \cong \mathcal{E}{\textit{nd}}(E^\vee )$ -module. The following is a consequence of Morita equivalence [Reference CăldăraruCal00, Reference LieblichLie07].

This discussion also works for the associated $\mu _r$ -gerbe, instead of the $\mathbb{G}_m$ -gerbe.

In summary, for a Brauer–Severi variety $Y \to X$ with corresponding Azumaya algebra $\mathcal{A}$ on $X$ with corresponding $\mathbb{G}_m$ -gerbe $\mathcal{G}$ , we have the three equivalent categories

\begin{align*} {\mathrm {Coh}}(X,Y), \quad {\mathrm {Coh}}(X,\mathcal {A}), \quad {\mathrm {Coh}}(\mathcal {G})_1, \end{align*}

which in particular shows that (up to equivalence) the category only depends on the gerbe $\mathcal{G}$ . In the next section, we view the $\mathbb{G}_m$ -gerbe $\mathcal{G}$ as a Brauer class and establish in a different way that ${{\rm Coh}}(X,\mathcal{A})$ only depends on this Brauer class (Example 2.25). In view of this discussion, it seems reasonable to call any of these equivalent categories the category of twisted sheaves.

2.3 Brauer group

The material of the previous sections is closely related to the Brauer group. We review its most important properties.

The fact that the inverse is given by the opposite algebra follows from an alternative definition of Azumaya algebras as coherent $\mathcal{O}_X$ -algebras $\mathcal{A}$ , which are locally free, and for which

(7) \begin{align} \mathcal{A} \otimes \mathcal{A}^{{\rm op}} \to \mathcal{E}{\textit{nd}}(\mathcal{A}), \quad a \otimes a^{\prime} \mapsto (x \mapsto axa^{\prime}) \end{align}

is an isomorphism.

From Proposition 2.21 one can see that the category of left $\mathcal{A}$ -modules only depends on the Brauer class of $\mathcal{A}$ . One can also see this explicitly as follows.

Example 2.25. For an Azumaya algebra $\mathcal{A}$ on $X$ and locally free sheaf $E$ of finite rank on $X$ , we want to show that $\mathcal{A}$ and $\mathcal{A} \otimes \mathcal{E}nd(E)$ have equivalent categories of left modules. This follows from Morita theory [Reference LamLam99, §18D]. Indeed, $\mathcal{A} \otimes \mathcal{E}nd(E)$ is a $(\mathcal{A}, \mathcal{A}\otimes \mathcal{E}{\textit{nd}}(E))$ bimodule, which is locally free both as $\mathcal{A}$ - and $\mathcal{A} \otimes \mathcal{E}{\textit{nd}}(E)$ -module. Explicitly, the functor

\begin{align*} \mathrm {Coh}(X,\mathcal {A}) \to \mathrm {Coh}(X,\mathcal {A} \otimes \mathcal {E}nd(E)), \quad F \mapsto F \otimes E, \end{align*}

is an equivalence.

One might wonder if the converse is true: if $\mathcal{A}$ and $\mathcal{A}^{\prime}$ are such that their categories of modules are equivalent, are they also Brauer equivalent? Căldăraru’s conjecture, now a theorem [Reference AntieauAnt16], says that this is almost the case.

3.1 Chern characters

From now on we will in addition assume that $\dim (X)=2$ , i.e. $X$ is a surface, and that $H_1(X, \mathbb{Z})$ is torsion free (in the complex analytic topology). By Poincaré duality and the universal coefficient theorem, this implies that all groups $H_i(X,\mathbb{Z})$ , $H^i(X,\mathbb{Z})$ are torsion free.

In the next lemma, we use the assumption that $H_1(X, \mathbb{Z})$ is torsion free to construct lifts of Stiefel–Whitney classes.

Lemma 3.1. There is an isomorphism $H^2(X, \mathbb{Z})/rH^2(X, \mathbb{Z}) \cong H^2(X, \mu _r)$ induced by the long exact sequence associated with the short exact sequence

\begin{align*} 0 \to \mathbb {Z} \stackrel {r \cdot }{\to } \mathbb {Z} \to \mathbb {Z}/r\mathbb {Z} \to 0, \end{align*}

where $\mathbb{Z} / r\mathbb{Z} \cong \mu _r$ . In other words, every $w \in H^2(X, \mu _r)$ is represented by a $\xi \in H^2(X, \mathbb{Z})$ , which is unique up to translation by multiples of $r$ .

Another important consequence of the assumption $H_1(X,\mathbb{Z})_{{\rm tor}} = 0$ is that for any Brauer–Severi variety $\pi : Y \to X$ , the map

\begin{align*} \pi ^* : H^*(X,\mathbb {Z}) \hookrightarrow H^*(Y,\mathbb {Z}) \end{align*}

is an inclusion [Reference YoshiokaYos06, Lemma 1.6].

The lifts of Lemma 3.1 are used to define the Chern character of twisted sheaves. This definition is due to Yoshioka [Reference YoshiokaYos06]. On surfaces this gives the correct answer (due to Proposition 3.3, and the discussion at the end of this section), but for higher-dimensional varieties another definition is required.

Definition 3.2. If $\mathcal{A}$ is an Azumaya algebra on a surface $X$ , choose a representing element $\xi \in H^2(X,\mathbb{Z})$ for $w(\mathcal{A}) \in H^2(X,\mu _r)$ (Lemma 3.1). If $F$ is a left $\mathcal{A}$ -module, we define

\begin{align*} {\mathrm {ch}}_{\mathcal {A}}(F) := \frac {{\mathrm {ch}}(F)}{\sqrt {{\mathrm {ch}}(\mathcal {A})}} \quad \text {and} \quad \mathrm {ch}^\xi _{\mathcal {A}}(F) := e^{\xi /r}{\mathrm {ch}}_{\mathcal {A}}(F). \end{align*}

Here on the right we take the Chern character of coherent $\mathcal{O}_X$ -modules. Equivalently (via Proposition 2.18), if $\pi : Y \to X$ is the Brauer–Severi variety corresponding to $\mathcal{A} = \pi _*(\mathcal{E}{\textit{nd}(G^\vee )})$ and $F$ is a $Y$ -sheaf, we define

\begin{align*} {\mathrm {ch}}_G(F) := \frac {{\mathrm {ch}}(\pi _*(F \otimes G^\vee ))}{\sqrt {{\mathrm {ch}}(\pi _*(G \otimes G^\vee ))}}, \quad \mathrm {ch}^\xi _{G}(F) := e^{\xi /r} {\mathrm {ch}}_G(F), \end{align*}

where $G$ is the sheaf defined in Lemma 2.4. We view these Chern characters as elements of $H^*(X,\mathbb{Q})$ . The component of ${{\rm ch}}_G(F)$ in $H^0(X,\mathbb{Z})$ equals ${{\rm rk}}(F)$ .

We refer to the factor $e^{\xi /r}$ as the Huybrechts–Stellari twist introduced in [Reference Huybrechts and StellariHS05]. The class $\xi /r \in H^2(X,\mathbb{Q})$ is called a rational $B$ -field. The Huybrechts–Stellari twist has the desirable feature that ${\rm ch}^\xi _{\mathcal{A}}(F)$ has the following integrality property shown in [Reference YoshiokaYos06] for K3 surfaces (but it holds for arbitrary surfaces with $H_1(X,\mathbb{Z})$ torsion free [Reference Jiang and KoolJK21]).

Proposition 3.3. Let $\mathcal{A}$ be an Azumaya algebra on a surface $X$ . For any left $\mathcal{A}$ -module $F$ , choose a representing element $\xi$ for $w(\mathcal{A}) \in H^2(X,\mu _r)$ and write

\begin{align*} {\mathrm {ch}}_{\mathcal {A}}^\xi (F) = (s, c_1, \tfrac {1}{2} c_1^2 - c_2) \in H^*(X,\mathbb {Q}). \end{align*}

Then $c_1 \in H^2(X,\mathbb{Z})$ and $c_2 \in H^4(X,\mathbb{Z}) \cong \mathbb{Z}$ .

We wish to provide more motivation for Definition 3.2. Suppose that $\mathcal{A} = \mathcal{E}{\textit{nd}}(E^\vee )$ is a trivial Azumaya algebra. Then every left $\mathcal{A}$ -module $M$ is of the form $F \otimes E^\vee$ for $F$ a coherent sheaf on $X$ by Example 2.25. One might expect that ${{\rm ch}}_{\mathcal{A}}^{\xi }(M) ={{\rm ch}}(F)$ . And indeed, this can be verified: on a surface $X$ , we have

(8) \begin{align} {{\rm ch}}(E) \cdot e^{-2c_1(E)/r} ={{\rm ch}}(E^\vee ). \end{align}

Keeping in mind that $\xi = c_1(E)$ represents $w$ , see (6), we can rewrite this as

\begin{align*} {\mathrm {ch}}(F)= \frac {{\mathrm {ch}}(F \otimes E^\vee )}{\sqrt {{\mathrm {ch}}(E^\vee )^2}} = \frac {{\mathrm {ch}}(F \otimes E^\vee )}{\sqrt {{\mathrm {ch}}(\mathcal {E}{\textit {nd}}(E^\vee )) e^{-2\xi /r}}} = {\mathrm {ch}}_{\mathcal {A}}^\xi (M). \end{align*}

There is an alternative perspective. We can view our surface as a complex manifold and forget the complex structure; what remains is a topological space. There is a notion of topological Azumaya algebras on this space. Their classes take values in the topological Brauer group, which is the torsion part of $H^3(X, \mathbb{Z})$ , which is zero in our case because we assume $H_1(X,\mathbb{Z})_{{\rm tor}} = 0$ . Hence, any topological Azumaya algebra is trivialized by a topological vector bundle, and this vector bundle is unique up to tensoring by a topological line bundle (that is, a class in $H^2(X, \mathbb{Z})$ ). One can use this to define Chern classes, and a similar calculation shows that they are the same as the previous definition. For details on topological Azumaya algebras, we refer the reader to [Reference Antieau and WilliamsAW13]. Similar ideas to what we have described also appear in [Reference HeinlothHei05, Reference YoshiokaYos06].

3.2 Moduli spaces

Let $(X,H)$ be a smooth polarized surface and suppose $H_1(X,\mathbb{Z})_{{\rm tor}} = 0$ . Let $\pi : Y \to X$ be a Brauer–Severi variety of degree $r$ with corresponding Azumaya algebra $\mathcal{A}$ . On the one hand, one can consider moduli spaces of stable $Y$ -sheaves on $X$ constructed by Yoshioka [Reference YoshiokaYos06]. On the other hand, one can consider moduli spaces of generically simple torsion free left $\mathcal{A}$ -modules constructed by Hoffmann and Stuhler [Reference Hoffmann and StuhlerHS05]. We recall the main results on these moduli spaces. Since ${\rm Coh}(X,Y)$ and ${\rm Coh}(X,\mathcal{A})$ are equivalent, one expects these moduli spaces to be isomorphic as was shown by Reede [Reference ReedeRee18].

We first recall Yoshioka’s result [Reference YoshiokaYos06]. Let $G$ be the unique (up to scaling) non-trivial extension of $T_{Y/X}$ by $\mathcal{O}_Y$ of Lemma 2.4. For a $Y$ -sheaf $F$ , its twisted Hilbert polynomial is defined by [Reference YoshiokaYos06]

\begin{align*} \chi (G,F \otimes \pi ^* \mathcal {O}_X(mH)) = \chi (X,\pi _*(F \otimes G^\vee )(mH)). \end{align*}

Suppose $F$ is torsion free. Then the above polynomial has degree 2 and we denote its leading term by ${1 \over 2}a_2^G(F) m^2$ . One defines $F$ as semistable (with respect to $H$ ) if

\begin{align*} \frac {\chi (\pi _*(F^{\prime} \otimes G^\vee )(mH))}{a_2^G(F^{\prime})} \leq \frac {\chi (\pi _*(F \otimes G^\vee )(mH))}{a_2^G(F)}, \end{align*}

for all $Y$ -subsheaves $0 \neq F^{\prime} \subsetneq F$ . Stability is defined analogously with $\leq$ replaced by $\lt$ .

We sometimes prefer fixing Chern classes instead of Chern characters, in which case we write $M_Y^{H,ss}(s,c_1,c_2)$ and $M_Y^{H}(s,c_1,c_2)$ . Representing the Stiefel–Whitney class $w(Y)$ by $\xi \in H^2(X,\mu _r)$ , we sometimes prefer to fix ${\rm ch}_G^\xi (F) = {\rm ch} = (s, c_1, {1 \over 2} c_1^2 - c_2)$ , in which case we denote the corresponding moduli spaces by

\begin{align*} M_{Y,\xi /r}^{H,ss}(\mathrm {ch}), \quad M_{Y,\xi /r}^{H}(\mathrm {ch}), \quad M_{Y,\xi /r}^{H,ss}(s,c_1,c_2), \quad M_{Y,\xi /r}^{H}(s,c_1,c_2). \end{align*}

Then we have $s \in \mathbb{Z}$ , $c_1 \in H^2(X,\mathbb{Z})$ , and $c_2 \in \mathbb{Z}$ (Proposition 3.3).

We now recall Hoffmann and Stuhler’s result. They consider left $\mathcal{A}$ -modules $F$ which are torsion free (as $\mathcal{O}_X$ -module) and generically simple, i.e. over the generic point $\eta \in X$ the $\mathcal{A}_\eta$ -module $F_\eta$ is simple. By Wedderburn’s theorem, over the generic point $\mathcal{A}_\eta \cong M_n(D)$ for some division algebra $D$ over $\mathbb{C}(X)$ and some $n \in \mathbb{Z}_{\gt 0}$ , and simplicity means $F_\eta \cong D^{\oplus n}$ (with $\mathcal{A}_\eta$ -module structure induced by matrix multiplication). One crucial observation is that the algebra of $\mathcal{A}$ -endomorphisms ${{\rm End}} _{\mathcal{A}}(F)$ is a finite-dimensional $\mathbb{C}$ -algebra, which is moreover a division ring because we have an embedding [Reference Hoffmann and StuhlerHS05]

\begin{align*} {\mathrm {End}} _{\mathcal {A}}(F) \hookrightarrow {\mathrm {End}} _{\mathcal {A}_\eta }(F_\eta ) \cong D^{\mathrm {op}}, \end{align*}

hence ${{\rm End}} _{\mathcal{A}}(F) = \mathbb{C}$ . Therefore, we do not have non-trivial automorphisms. In fact, no notion of stability is required.

The property of generic simplicity implies $s = {\rm deg}(D) := \sqrt{{\rm dim}_{\mathbb{C}(X)}(D)}$ (and in particular $F$ has rank $ns^2$ as $\mathcal{O}_X$ -module). Therefore, we may suppress rank $s$ from the notation. Also recall that $\mathcal{A}$ has degree $r$ , so $r,n,s$ are related by $r = ns$ . We sometimes prefer fixing Chern classes instead of Chern characters, in which case we write $M_{\mathcal{A}}(c_1,c_2)$ . Moreover, sometimes we fix the twisted Chern character ${\rm ch}^\xi _{\mathcal{A}}(F) = {\rm ch} = (s, c_1, {1 \over 2} c_1^2 - c_2)$ , in which case we denote the corresponding moduli space by

\begin{align*} M_{\mathcal {A},\xi /r}(\mathrm {ch}), \quad M_{\mathcal {A},\xi /r}(c_1,c_2). \end{align*}

In the setting of Azumaya algebras, one often only considers the case $n=1$ , i.e. $\mathcal{A}_\eta \cong D$ . The reason is that one can replace $\mathcal{A}$ by a Brauer equivalent Azumaya algebra with this property [Reference ReedeRee18, Remark 2.1] and replacing $\mathcal{A}$ by a Brauer equivalent Azumaya algebra does not change the category of left modules (Example 2.25).

Then the degree $r$ of the Azumaya algebra $\mathcal{A}$ is equal to its index which is defined as the degree of $D$ . By the period-index theorem [Reference de JongdJo04], this is also the period of $\mathcal{A}$ which is defined as the order of the Brauer class $\alpha (\mathcal{A})$ .Footnote ¹¹ For $n=1$ , we have the following result.

Theorem 3.7. (Reede). The equivalence of categories of Proposition 2.18 induces an isomorphism of moduli spaces

\begin{align*} M_{Y}^{H}(r,c_1,c_2) \cong M_{\mathcal {A}}(c_1,c_2). \end{align*}

In particular, all rank $r$ torsion free $Y$ -sheaves are automatically stable and the space $M_{Y}^{H}(r,c_1,c_2)$ is projective and independent of $H$ .

In fact, Reede identifies the moduli functors, not just the coarse spaces.

Still working under the assumption $\mathcal{A}_\eta \cong D$ , Hoffmann and Stuhler consider the locus

\begin{align*} M_{\mathcal {A}}^{\mathrm {lf}}(c_1,c_2) \subset M_{\mathcal {A}}(c_1,c_2) \end{align*}

of locally free $\mathcal{A}$ -modules of rank 1 (as $\mathcal{A}$ -modules). Note that ${\rm Pic}(X)$ acts on the union of these moduli spaces over all $c_1, c_2$ by $F \mapsto F \otimes L$ .

Proposition 3.8. (Hoffmann–Stuhler). The map $F \to \mathcal{E}{\textit{nd}}_{\mathcal{A}}(F)^{{\rm op}}$ gives a bijection between the closed points of

\begin{align*} \bigsqcup _{c_1,c_2} M_{\mathcal {A}}^{\mathrm {lf}}(c_1,c_2) \Big / \mathrm {Pic}(X) \end{align*}

and the set of isomorphism classes of Azumaya algebras $\mathcal{B}$ on $X$ satisfying $\mathcal{B}_\eta \cong D$ . Under this bijection $c_2(\mathcal{B}) =2rc_2 - (r-1) c_1^2$ .

Proof. This is [Reference Hoffmann and StuhlerHS05, Proposition 4.1], except for the calculation of $c_2(\mathcal{B})$ with $\mathcal{B} = \mathcal{E}{\textit{nd}}_{\mathcal{A}}(F)^{{\rm op}}$ where $F$ is a locally free $\mathcal{A}$ -modules of rank 1 (as $\mathcal{A}$ -module). The natural map

\begin{align*} \mathcal {A} \otimes _{\mathcal {O}_X} \mathcal {E}{\textit {nd}}_{\mathcal {A}}(F) \to \mathcal {E}{\textit {nd}}(F) \end{align*}

is an isomorphism. Since $c_1(\mathcal{A}) = 0$ , we obtain

(9) \begin{align}{{\rm ch}}(\mathcal{B}) ={{\rm ch}}(\mathcal{E}{\textit{nd}}_{\mathcal{A}}(F)) = {{{\rm ch}}(\mathcal{E}{\textit{nd}}(F)) \over {{\rm ch}}(\mathcal{A})} = {{{\rm ch}}(F) \over \sqrt{{{\rm ch}}(\mathcal{A})}} \Bigg ( {{{\rm ch}}(F) \over \sqrt{{{\rm ch}}(\mathcal{A}})} \Bigg )^\vee = r^2 + (r-1)c_1^2 -2rc_2, \end{align}

from which the result follows.

Corollary 3.9. If $M_{\mathcal{A}}(c_1,c_2) \neq \varnothing$ , then there exists an Azumaya algebra $\mathcal{B}$ on $X$ satisfying $\mathcal{B}_\eta \cong D$ and

\begin{align*} c_2(\mathcal {B}) \leq 2rc_2 - (r-1) c_1^2. \end{align*}

Proof. Taking an element $F$ in $M_{\mathcal{A}}(c_1,c_2)$ , its double dual (as $\mathcal{O}_X$ -module) $F^{**} = \mathcal{H}{\textit{om}}(\mathcal{H}{\textit{om}}(F,\mathcal{O}_X),\mathcal{O}_X)$ has a natural left $\mathcal{A}$ -module structure and is locally free (as $\mathcal{O}_X$ -module). Consider the short exact sequence of $\mathcal{O}_X$ -modules

\begin{align*} 0 \to F \to F^{**} \to Q \to 0, \end{align*}

induced by the natural inclusion $F \hookrightarrow F^{**}$ . Then $Q$ is 0-dimensional, so ${{\rm rk}}(F^{**}) ={{\rm rk}}(F) = r^2$ (implying $F^{**}$ is locally free of rank 1 as $\mathcal{A}$ -module by Proposition 2.21), $c_1(F^{**}) = c_1(F)$ , and $c_2(F^{**}) = c_2(F) + c_2(Q) \leq c_2(F)$ . Taking $\mathcal{B} = \mathcal{E}{\textit{nd}}_{\mathcal{A}}(F^{**})^{{\rm op}}$ , the result follows from Proposition 3.8.

In § 5.3, we present a method to detect the condition ‘ $M_{\mathcal{A}}(c_1,c_2) \neq \varnothing$ ’ by showing that certain intersection numbers on the moduli space are non-zero.

4.1 Obstruction theory

Let $(X,H)$ be a smooth polarized surface satisfying $H_1(X,\mathbb{Z}) = 0$ . Let $Y \to X$ be a Brauer–Severi variety of degree $r$ and consider the moduli spaces $M:=M_{Y}^{H}({\rm ch})$ of the previous section.Footnote ¹⁴ Denote by $\pi _M : Y \times M \rightarrow M$ the projection. Although a universal sheaf $\mathcal{E}$ may not exist globally on $Y \times M$ , the complex

\begin{align*} R \mathcal {H}{\textit {om}}_{\pi _M}(\mathcal {E},\mathcal {E}) := R\pi _{M*} R\mathcal {H}{\textit {om}}(\mathcal {E},\mathcal {E}) \end{align*}

exists globally on $Y \times M$ [Reference CăldăraruCal00, Reference Huybrechts and LehnHL10]. Put differently, $\mathcal{E}$ exists as a twisted sheaf on $Y \times M$ . Denote the truncated cotangent complex of $M$ by $\mathbb{L}_M = \tau ^{\geq -1} L_M$ . The following result is well known in the untwisted case [Reference Huybrechts and ThomasHT09] and shown in the twisted case in [Reference Jiang and KoolJK21].

Proposition 4.1. Fix ${{\rm ch}} = (s,c_1,{1 \over 2} c_1^2 - c_2) \in H^*(X,\mathbb{Q})$ . Then the moduli space $M:=M_{Y}^{H}({\rm ch})$ has a perfect obstruction theory

\begin{align*}\mathbb {E} := (T_M^{\mathrm {vir}})^\vee := (R \mathcal {H}{\textit {om}}_{\pi _M}(\mathcal {E},\mathcal {E})_0[1])^\vee \rightarrow \mathbb {L}_M \end{align*}

of virtual dimension

(10) \begin{align} {\rm vd}({\rm ch}) := {\rm vd}(s,c_1,c_2) := 2sc_2 - (s-1)c_1^2 - (s^2-1)\chi (\mathcal{O}_X). \end{align}

By the work of Behrend and Fantechi [Reference Behrend and FantechiBF97], we therefore obtain a virtual fundamental class of degree ${\rm vd}(s,c_1,c_2)$ in the Chow group of $M$

\begin{align*} [M]^{\mathrm {vir}} \in A_{\mathrm {vd}(s,c_1,c_2)}(M). \end{align*}

We provide the following two additional perspectives on this obstruction theory.

Remark 4.2. Let $\mathcal{G} \to X$ be the $\mu _r$ -gerbe corresponding to $w(Y) \in H^2(X,\mu _r)$ . Then the analogue of this proposition was shown for the moduli stack ${\bf Tw}_{\mathcal{G}}^{s}(r,0,c_2)$ of twisted sheaves on $\mathcal{G}$ by Lieblich [Reference LieblichLie09, Proposition 6.5.1.1]. Another way to view this proposition is as follows. Let $\boldsymbol{M}$ be the derived stack of simple sheaves with fixed determinant on $Y$ . Then its classical truncation $M^{{\rm cl}}$ has an obstruction theory

\begin{align*} \mathbb {E} := L_{\boldsymbol {M}}|_{M^{\mathrm {cl}}} = (R \mathcal {H}{\textit {om}}_{\pi _{M^{\mathrm {cl}}}}(\mathcal {E},\mathcal {E})_0[1])^\vee \to L_{M^{\mathrm {cl}}}. \end{align*}

Yoshioka proved that the locus of $Y$ -sheaves is open in $M^{{\rm cl}}$ [Reference YoshiokaYos06, Lemma 1.6.5], so we can restrict $\mathbb{E} \to L_{M^{{\rm cl}}}$ to this open locus (alhough we are now on a stack, which is a $\mathbb{G}_m$ -gerbe over the moduli scheme in Proposition 4.1).

We are interested in the relative situation. Suppose $f : \mathcal{X} \to B$ is a smooth projective morphism of relative dimension 2 with connected fibres over a smooth connected variety $B$ . We assume that one fibre (and hence all fibres) $\mathcal{X}_b$ satisfies $H_1(\mathcal{X}_b,\mathbb{Z}) = 0$ . Suppose $\pi : \mathcal{Y} \to \mathcal{X}$ is a smooth projective morphism of relative dimension $r-1$ such that the fibres over closed points $b \in B$ are Brauer–Severi varieties $\mathcal{Y}_b \to \mathcal{X}_b$ . Moreover, we denote by $G_b$ the unique (up to scaling) non-trivial extension of $T_{\mathcal{Y}_b/\mathcal{X}_b}$ by $\mathcal{O}_{\mathcal{Y}_b}$ (Lemma 2.4).

Consider the Hodge bundles (with respect to the Zariski topology)

\begin{align*} \mathcal {H}^{2p}_{\mathrm {dR}} := \mathcal {H}_{\mathrm {dR}}^{2p}(\mathcal {X} / B) := R^{2p} f_* \Omega _{\mathcal {X} / B}^{\bullet }, \end{align*}

where $\Omega _{\mathcal{X} / B}^{\bullet }$ is the algebraic de Rham complex. Since the family $\mathcal{X} \to B$ is fixed, we suppress it from the notation. Recall that Hodge bundles behave well with respect to base change and the fibre of $\mathcal{H}^{2p}_{{\rm dR}}$ over a closed point $b \in B$ is

\begin{align*} H^{2p}(\mathcal {X}_b,\Omega _{\mathcal {X}_b}^\bullet ) \cong H^{2p}(\mathcal {X}_b,\mathbb {C}), \end{align*}

where on the right-hand side, we consider $\mathcal{X}_b$ with the complex analytic topology. We fix a flat section with respect to the Gauß–Manin connection

\begin{align*} \widetilde {v} = (\widetilde {v}_0, \widetilde {v}_1, \widetilde {v}_2) \in \bigoplus _{p=0}^{2} \Gamma (B, \mathcal {H}^{2p}_{\mathrm {dR}}). \end{align*}

Fix a family of polarizations $\mathcal{H}$ on $\mathcal{X}$ . Denote by

\begin{align*} M_{\mathcal {Y} / B} := M_{\mathcal {Y} / B}^{\mathcal {H}} (\widetilde {v}) \end{align*}

the moduli space parametrizing $\mathcal{H}_b$ -stable $\mathcal{Y}_b$ -sheaves $F$ with Chern character ${{\rm ch}}_{G_b}(F) = \widetilde{v}_b$ , for some closed point $b \in B$ . This is the relative version over $B$ of the moduli spaces constructed by Yoshioka [Reference YoshiokaYos06]. Note that for any closed point $b \in B$ , we have a Cartesian diagram

where $M_b := M_{\mathcal{Y}_b}^{\mathcal{H}_b} (\widetilde{v}_b)$ .

Remark 4.3. In order for these moduli spaces to be non-empty, we must have

(11) \begin{align} \widetilde{v}_b \in \bigoplus _{p=0}^{2} H^{p,p}(\mathcal{X}_b), \end{align}

for some closed point $b \in B$ . Suppose $B$ is quasi-projective. Then Deligne’s invariant cycle theorem implies that (11) holds for all closed points $b \in B$ [Reference EduardoCat14, Proposition 11.3.5].

By [Reference Huybrechts and ThomasHT09, Theorem 4.1], we have a relative obstruction theory

\begin{align*} \phi : \mathbb {E}_{\mathcal {Y}/B} \to \mathbb {L}_{M_{\mathcal {Y}/B}/B}. \end{align*}

For each closed point $b \in B$ , consider the inclusion $j_b : \mathcal{X}_b \hookrightarrow \mathcal{X}$ , then we obtain an induced map

\begin{align*} \phi _b : \mathbb {E}_b := Lj_b^* \mathbb {E}_{\mathcal {Y}/B} \to Lj_b^* \mathbb {L}_{M_{\mathcal {Y}/B}/B} \to \mathbb {L}_{M_b}, \end{align*}

which is the perfect obstruction theory of Proposition 4.1. Since the base $B$ is connected, the topological number

\begin{align*} \mathrm {vd} = \mathrm {vd}(\widetilde {v}_b) \in \mathbb {Z}_{\geq 0} \end{align*}

does not depend on the closed point $b \in B$ . We obtain a cycle class [Reference Huybrechts and ThomasHT09, Corollary 4.3]

\begin{align*} [M_{\mathcal {Y}/B}]^{\mathrm {vir}} \in A_{\mathrm {vd}+ \dim (B)}(M_{\mathcal {Y}/B}), \end{align*}

such that for all closed points $b \in B$ we have

(12) \begin{align} i_b^! [M_{\mathcal{Y}/B}]^{{\rm vir}} = [M_b]^{{\rm vir}} \in A_{{\rm vd}}(M_b), \end{align}

where $i_b^! : A_*(M_{\mathcal{Y}/B}) \to A_*(M_{b})$ is the refined Gysin pull-back [Reference FultonFul98] for the Cartesian diagram (4.1).

4.2 The ${\rm SL}_r$ and ${\rm PGL}_r$ generating function

Let $R$ be a commutative $\mathbb{Q}$ -algebra e.g. $\mathbb{Q}$ or a polynomial algebra over $\mathbb{Q}$ . Fix $v \in \mathbb{Z}_{\geq 0}$ , and $\boldsymbol{a} = (a_1, \ldots, a_N) \in \{1,2\}^N$ for some $N \in \mathbb{Z}_{\geq 0}$ . Let $(\underline{\alpha _0},\underline{\alpha _1},\ldots, \underline{\alpha _N})$ be a list of variables and let $\underline{r}$ be a further variable. Let $\mathsf{P}_{v,\boldsymbol{a}}$ be a formal power series, with coefficients in $R(\underline{r})$ , in the following formal symbols:

\begin{align*} & \pi _{M*}\Big ( \pi _X^* \underline{\alpha _i} \cap{{\rm ch}}_{k}( \underline{\mathcal{E}} \otimes \det (\underline{\mathcal{E}})^{-{1 \over \underline{r}}})\Big ), \quad \pi _{M*}\Big ( \pi _X^* \underline{\alpha _i c_1(X)} \cap{{\rm ch}}_{k}( \underline{\mathcal{E}} \otimes \det (\underline{\mathcal{E}})^{-{1 \over \underline{r}}})\Big ), \\ & \pi _{M*}\Big ( \pi _X^* \underline{\alpha _i c_2(X)} \cap{{\rm ch}}_{k}( \underline{\mathcal{E}} \otimes \det (\underline{\mathcal{E}})^{- {1 \over \underline{r}}})\Big ), \quad \pi _{M*}\Big ( \pi _X^* \underline{\alpha _i c_1(X)^2} \cap{{\rm ch}}_{k}( \underline{\mathcal{E}} \otimes \det (\underline{\mathcal{E}})^{-{1 \over \underline{r}}})\Big ), \\ & c_j(R\mathcal{H}{\textit{om}}_{\pi _M}(\underline{\mathcal{E}}, \underline{\mathcal{E}})[1]), \end{align*}

where $i=0,\ldots, N$ and $k,j \in \mathbb{Z}_{\geq 0}$ , and $\underline{\alpha _i}, \underline{\alpha _i c_1(X)}, \ldots, \underline{\mathcal{E}}$ are viewed as formal variables. Define degrees $\deg \, c_j(\cdot ) = j$ and

\begin{align*} \deg \,\pi _{M*}\Big ( \pi _X^* \underline{\alpha _i} \cap{{\rm ch}}_{k}( \underline{\mathcal{E}} \otimes \det (\underline{\mathcal{E}})^{- {1 \over \underline{r}}})\Big ) & = a_i + k-2, \\ \deg \,\pi _{M*}\Big ( \pi _X^* \underline{\alpha _i c_1(X)} \cap{{\rm ch}}_{k}( \underline{\mathcal{E}} \otimes \det (\underline{\mathcal{E}})^{- {1 \over \underline{r}}})\Big ) & = a_i + k - 1, \\ \deg \,\pi _{M*}\Big ( \pi _X^* \underline{\alpha _i c_2(X)} \cap{{\rm ch}}_{k}( \underline{\mathcal{E}} \otimes \det (\underline{\mathcal{E}})^{- {1 \over \underline{r}}})\Big ) & = a_i + k, \\ \deg \,\pi _{M*}\Big ( \pi _X^* \underline{\alpha _i c_1(X)^2} \cap{{\rm ch}}_{k}( \underline{\mathcal{E}} \otimes \det (\underline{\mathcal{E}})^{-{1 \over \underline{r}}})\Big ) & = a_i + k, \end{align*}

where $i=0,\ldots, N$ and we take $a_0:=0$ . We assume that the formal power series $\mathsf{P}_{v,\boldsymbol{a}}$ has only finitely many terms in each degree. We refer to $\mathsf{P}_{v,\boldsymbol{a}}$ as a formal insertion. The reader who is only interested in virtual Euler characteristics can take $\mathsf{P}_v = c_v(R\mathcal{H}{\textit{om}}_{\pi _M}(\underline{\mathcal{E}}, \underline{\mathcal{E}})[1])$ .

Keeping $\boldsymbol{a}$ , $\underline{r}$ fixed as above, we now consider a sequence of formal insertions $\mathsf{P} := \{\mathsf{P}_{v,\boldsymbol{a}}\}_{v \geq 0}$ .Footnote ¹⁵ Let $r \gt 1$ , let $(X,H)$ be a smooth polarized surface satisfying $H_1(X,\mathbb{Z}) = 0$ , and $\boldsymbol{\alpha } = (\alpha _1, \ldots, \alpha _N) \in H^{*}(X,\mathbb{Q})^N$ algebraic classes with $\deg (\alpha _i) = a_i$ for all $i=1, \ldots, N$ .

${\rm SL}_r$ -invariants.

As in the introduction, denote by $M:=M_X^H(r,c_1,c_2)$ the Gieseker–Maruyama–Simpson moduli space of rank $r$ Gieseker $H$ -stable sheaves on $X$ with Chern classes $c_1,c_2$ [Reference Huybrechts and LehnHL10]. Suppose, for the moment, that there exists a universal sheaf $\mathcal{E}$ on $X \times M$ such that the line bundle $\det (\mathcal{E})$ has an $r$ th root. We drop these two assumptions in Remark 4.7. Take

\begin{align*} v := {\mathrm {rk}}(R\mathcal {H}{\textit {om}}_{\pi _M}(\mathcal {E},\mathcal {E})[1]) + \chi (\mathcal {O}_X). \end{align*}

Then the evaluation $\mathsf{P}_{v,\boldsymbol{a}}(r,X,\boldsymbol{\alpha };M,\mathcal{E})$ is defined as the cohomology class on $M$ obtained from $\mathsf{P}_{v,\boldsymbol{a}}$ by substitutingFootnote ¹⁶

\begin{align*} & \underline{\alpha _0} = 1, \quad \underline{\alpha _0 c_1(X)} = c_1(X), \quad \underline{\alpha _0 c_2(X)} = c_2(X), \quad \underline{\alpha _0 c_1(X)^2} = c_1(X)^2, \quad \underline{\mathcal{E}} = \mathcal{E}, \\ & \underline{r} = r, \quad \underline{\alpha _i} = \alpha _i, \quad \underline{\alpha _i c_1(X)} = \alpha _i c_1(X), \quad \underline{\alpha _i c_2(X)} = \alpha _i c_2(X), \quad \underline{\alpha _i c_1(X)^2} = \alpha _i c_1(X)^2 \end{align*}

for all $i=1, \ldots, N$ , where $\pi _X : X \times M \to M$ , $\pi _M : X \times M \to M$ denote the projections.

Definition 4.5. Fix a first Chern class $c_1 \in H^2(X,\mathbb{Z})$ . Suppose there are no rank $r$ strictly Gieseker $H$ -semistable sheaves on $X$ with first Chern class $c_1$ (this is the case, for example, when $\gcd (r,c_1H)=1$ ). We define the ${\rm SL}_r$ generating function of $(X,H), c_1$ associated with formal insertions $\mathsf{P} := \{\mathsf{P}_{v,\boldsymbol{a}}\}_{v \geq 0}$ by

\begin{align*} \mathsf {Z}_{(X,H),c_1}^{\mathrm {SL}_r, \mathsf {P}}(q) := \sum _{c_2 \in \mathbb {Z}} q^{\frac {\mathrm {vd}(r,c_1,c_2)}{2r}} \int _{[M_X^H(r,c_1,c_2)]^{\mathrm {vir}}} \mathsf {P}_{\mathrm {vd}(r,c_1,c_2),\boldsymbol {a}}(r,X,\boldsymbol {\alpha };M_X^H(r,c_1,c_2),\mathcal {E}), \end{align*}

where ${\rm vd}(r,c_1,c_2)$ was defined in (10).

${\rm PGL}_r$ -invariants.

Fix a class $w \in H^2(X,\mu _r)$ . By Theorem 2.16, there exists a degree $r$ Brauer–Severi variety $Y \to X$ such that $w(Y) = w$ . Choose a representative $\xi \in H^2(X,\mathbb{Z})$ of $w$ , i.e. $w = [\xi ]$ (Lemma 3.1). As in § 3.2, denote by $M:=M_{Y,\xi /r}^H(r,\xi, c_2)$ Yoshioka’s moduli space of $H$ -stable $Y$ -sheaves $F$ satisfying

\begin{align*} {\mathrm {ch}}_G^{\xi }(F) = (r,\xi, \tfrac {1}{2} \xi^2 - c_2). \end{align*}

Note that we have ‘switched on’ the $B$ -field so that $c_2 \in \mathbb{Z}$ (Proposition 3.3). Also note that we choose the rank of $F$ equal to the degree of $Y$ and ‘twisted’ first Chern class equal to $\xi$ .Footnote ¹⁷ Suppose, for the moment, that there exists a universal sheaf $\mathcal{E}$ on $Y \times M$ such that the line bundle $\det (\mathcal{E})$ has an $r$ th root. We drop these two assumptions in Remark 4.7. Take

\begin{align*} v := {\mathrm {rk}}(R\mathcal {H}{\textit {om}}_{\pi _M}(\mathcal {E},\mathcal {E})[1]) + \chi (\mathcal {O}_X). \end{align*}

The evaluation $\mathsf{P}_{v,\boldsymbol{a}}(r,X,\boldsymbol{\alpha };M,\mathcal{E})$ is defined as the cohomology class on $M$ obtained from $\mathsf{P}_{v,\boldsymbol{a}}$ by substituting

\begin{align*} \underline {\mathcal {E}} \otimes \det (\underline {\mathcal {E}})^{-\frac {1}{\underline {r}}} = \pi _*(\mathcal {E} \otimes \det (\mathcal {E})^{-\frac {1}{r}}), \end{align*}

where $\pi$ is the base change of $\pi : Y \to X$ to $Y \times M$ , and furthermore by substituting

\begin{align*} & \underline{\alpha _0} = 1, \quad \underline{\alpha _0 c_1(X)} = c_1(X), \quad \underline{\alpha _0 c_2(X)} = c_2(X), \quad \underline{\alpha _0 c_1(X)^2} = c_1(X)^2, \\ & \underline{r} = r, \quad \underline{\alpha _i} = \alpha _i, \quad \underline{\alpha _i c_1(X)} = \alpha _i c_1(X), \quad \underline{\alpha _i c_2(X)} = \alpha _i c_2(X), \quad \underline{\alpha _i c_1(X)^2} = \alpha _i c_1(X)^2, \end{align*}

for all $i=1, \ldots, N$ , where $\pi _X : X \times M \to M$ , $\pi _M : X \times M \to M$ denote the projections.

Definition 4.6. Fix $w \in H^2(X,\mu _r)$ . Choose a degree $r$ Brauer–Severi variety $Y \to X$ with $w(Y) = w$ and choose a representative $\xi \in H^2(X,\mathbb{Z})$ of $w$ . Suppose there are no rank $r$ strictly $H$ -semistable $Y$ -sheaves $F$ with ${\rm ch}_{G}^{\xi }(F) = (r,\xi, {1 \over 2} \xi^2 - c_2)$ for any $c_2$ (this is the case, for example, when the Brauer class $o(w) \in H^2(X,\mathbb{G}_m)$ has order $r$ by Remark 3.5). We define the ${\rm PGL}_r$ generating function of $(X,H), w$ associated with formal insertions $\mathsf{P} := \{\mathsf{P}_{v,\boldsymbol{a}}\}_{v \geq 0}$ by

\begin{align*} \mathsf {Z}_{(X,H),w}^{\mathrm {PGL}_r, \mathsf {P}}(q) := \sum _{c_2 \in \mathbb {Z}} q^{\frac {\mathrm {vd}(r,\xi, c_2)}{2r}} \int _{[M_{Y,\xi /r}^H(r,\xi, c_2)]^{\mathrm {vir}}} \mathsf {P}_{\mathrm {vd}(r,\xi, c_2),\boldsymbol {a}}(r,X,\boldsymbol {\alpha };M_{Y,\xi /r}^H(r,\xi, c_2),\mathcal {E}), \end{align*}

where ${\rm vd}(r,\xi, c_2)$ was defined in (10). As we will see below (Proposition 4.9), this generating function does not depend on the choice of $Y, \xi$ on the right-hand side, justifying the notation.

Remark 4.7. In each of the above two settings, in general a universal sheaf $\mathcal{E}$ on $X \times M$ (respectively $Y \times M$ ) may only exist étale locally. However, note that the following complex and sheaf always exist globally by [Reference CăldăraruCal00, Theorem 2.2.4] (see also [Reference Huybrechts and LehnHL10, §10.2]):

\begin{align*} R\mathcal {H}{\textit {om}}_{\pi _M}(\mathcal {E},\mathcal {E}), \quad \mathcal {E}^{\otimes r} \otimes \det (\mathcal {E})^{-1}. \end{align*}

Note furthermore that for any class $V \in 1+K^0(X \times M)$ , we can define an $r$ th root operation

\begin{align*} \sqrt [r]{V} \in K^0(X \times M)_{\mathbb {Q}} \end{align*}

precisely as in [Reference Oh and ThomasOT23, Lemma 5.1] (which handles the $r=2$ case, but the arguments generalize). Then we set

\begin{align*} {\mathrm {ch}}(\mathcal {E} \otimes (\det (\mathcal {E}))^{-\frac {1}{r}}) := \mathrm {ch}\Big ( r\sqrt [r]{\frac {1}{r^r} \cdot \mathcal {E}^{\otimes r} \otimes \det (\mathcal {E})^{-1} } \Big ), \end{align*}

and similarly for ${{\rm ch}}(\pi _*(\mathcal{E} \otimes (\det (\mathcal{E}))^{-{1 \over r}}))$ for a degree $r$ Brauer–Severi variety $\pi : Y \to X$ .

We will now prove that the generating function $\mathsf{Z}^{{\rm PGL}_r, \mathsf{P}}_{(X,H), w}(q)$ does not depend on the choice of Brauer–Severi variety $Y$ with Stiefel–Whitney class $w$ , or representative $\xi$ of $w$ . Here we will make crucial use of [Reference YoshiokaYos06, Lemma 3.5], which establishes the relevant isomorphism of moduli spaces. We extend this result to also include an isomorphism of virtual tangent bundles. Furthermore, we prove [Reference YoshiokaYos06, Lemma 3.5] in a slightly different way, and we will provide more details and include some intermediate steps that are useful on their own. We first recall Yoshioka’s set-up.

Suppose $p_1 : Y_1 \to X$ and $p_2 : Y_2 \to X$ are two Brauer–Severi varieties with the same Brauer class $o(w(Y_1)) = o(w(Y_2))$ (we will later assume $Y_1$ , $Y_2$ have the same degree and $w(Y_1) = w(Y_2)$ ). Denote the projections $Y_1 \times _X Y_2 \to Y_1$ and $Y_1 \times _X Y_2 \to Y_2$ by $\pi _1$ and $\pi _2$ , respectively. Yoshioka shows that there exists a line bundle $L$ on $Y_1 \times _X Y_2$ such that the Fourier–Mukai transform

\begin{align*} \Xi : {\mathrm {Coh}}(X, Y_1) \to {\mathrm {Coh}}(X, Y_2), \quad \Xi (F) = \pi _{2*}(\pi _1^*F \otimes L) \end{align*}

defines an equivalence of categories [Reference YoshiokaYos06, Lemma 1.7]. Note that we can always replace $L$ by $L \otimes \pi _2^*p_2^*P$ with $P$ a line bundle on $X$ , something we will do later. We first record the following identities.

Lemma 4.8. Let $F$ be an $Y_1$ -sheaf. The canonical map

\begin{align*} \pi _2^*\Xi (F) = \pi _2^*\pi _{2*}(\pi _1^*F \otimes L) \to \pi _1^*F \otimes L \end{align*}

is an isomorphism, i.e. $\pi _1^*F \otimes L$ is globally generated relative to $\pi _2$ . If $F^{\prime}$ is another $Y_1$ -sheaf that is locally free, we have that

\begin{align*} \pi _2^*(\Xi (F^{\prime})^\vee \otimes \Xi (F)) \cong \pi _1^*((F^{\prime})^\vee \otimes F). \end{align*}

Proof. We can check the first statement étale locally, where the Brauer–Severi varieties are trivial. Then, the sheaves $\mathcal{O}_{Y_1}(1)$ and $\mathcal{O}_{Y_2}(1)$ exist. Yoshioka has already shown that $F(-1)$ is globally generated relative to $p_1$ . Then, by base change, $\pi _1^*(F(-1))$ is globally generated relative to $\pi _2$ , and also, $\pi _1^*(F(-1)) \otimes \pi _2^*Q$ is globally generated relative to $\pi _2$ for any line bundle $Q$ . The claim now follows from the construction of $L$ by Yoshioka: on this étale cover it is given by $\mathcal{O}_{Y_1}(-1) \boxtimes (\mathcal{O}_{Y_2}(1) \otimes p_2^*P)$ for $P$ any line bundle on $X$ .

The second statement directly follows from the first.

Proposition 4.9. The generating function $\mathsf{Z}_{(X,H),w}^{{\rm PGL}_r, \mathsf{P}}(q)$ does not depend on the choice of degree $r$ Brauer–Severi variety $Y \to X$ with $w(Y) = w$ or representative $\xi \in H^2(X,\mathbb{Z})$ of $w$ .

Proof. We start with the independence of $Y$ . Using the above notation, let $Y_1, Y_2$ be degree $r$ Brauer–Severi varieties on $X$ with $w(Y_1) = w(Y_2) = w$ . We will first show that $\Xi$ induces an isomorphism between moduli spaces, and then that it preserves the obstruction theory and the generating function.

We first want to prove that $\Xi$ preserves the Chern character, in order to have a chance at showing it preserves the moduli spaces. Denote by $G_i$ the unique (up to scaling) non-trivial extension of $T_{Y_i/X}$ by $\mathcal{O}_{Y_i}$ (Lemma 2.4). We consider the equality ${{\rm ch}}_{G_1}(F) ={{\rm ch}}_{G_2}(\Xi (F))$ for any $Y_1$ -sheaf $F$ of positive rank. It suffices to show this equality on $Y_1 \times _X Y_2$ modulo $H^{\geq 6}$ , since $Y_1 \times _X Y_2 \to X$ induces an injective map on cohomology. Using the fact that $F, G_1$ are $Y_1$ -sheaves and $\Xi (F), G_2$ are $Y_2$ -sheaves (Definition 2.17), it suffices to prove the following equality modulo $H^{\geq 6}$ :

\begin{align*} {\mathrm {ch}} \left ((\pi _1^* F \otimes \pi _1^*G_1^\vee )^{\otimes 2} \otimes \pi _2^*G_2 \otimes \pi _2^*G_2^\vee \right ) = {\mathrm {ch}} \left ( (\pi _2^*(\Xi F) \otimes \pi _2^*G_2^\vee )^{\otimes 2} \otimes \pi _1^*G_1 \otimes \pi _1^*G_1^\vee \right ). \end{align*}

By Lemma 4.8, this is equivalent to the following equation modulo $H^{\geq 6}$ :

\begin{align*}{{\rm ch}} & \left ( (\pi _1^* F)^{\otimes 2} \otimes \pi _1^*G_1^\vee \otimes \pi _2^*(\Xi G_1)^\vee \otimes \pi _2^*G_2 \otimes \pi _2^*G_2^\vee \right )\\ & ={{\rm ch}} \left ( (\pi _1^* F)^{\otimes 2} \otimes (\pi _2^*G_2^\vee )^{\otimes 2} \otimes \pi _2^*(\Xi G_1) \otimes \pi _1^*G_1^\vee \right ), \end{align*}

which can then be rearranged to give

\begin{align*} {\mathrm {ch}}( (\pi _1^*F)^{\otimes 2} \otimes \pi _1^*G_1^\vee \otimes \pi _2^*G_2^\vee ) \cdot \pi _2^*{\mathrm {ch}}( \Xi (G_1)^\vee \otimes G_2 - G_2^\vee \otimes \Xi (G_1))=0. \end{align*}

Thus, to obtain the Chern character equality, it suffices to show that ${{\rm ch}}(\Xi (G_1)^\vee \otimes G_2) ={{\rm ch}}(G_2^\vee \otimes \Xi (G_1))$ modulo $H^{\geq 6}(Y_2,\mathbb{Q})$ . Since, ${\rm ch}_0$ and ${\rm ch}_2$ already agree, we only have to worry about the $c_1$ . We show that we can replace $L$ by $L \otimes \pi _2^*p_2^*P$ for some $P \in {\rm Pic}(X)$ such that $c_1(G_2) = c_1(\Xi (G_1))$ , which proves what we want. Changing $L$ by $L \otimes \pi _2^* p_2^* P$ replaces $\Xi$ by $F \mapsto p_2^*P \otimes \Xi (F)$ , so, in particular, it replaces $c_1(\Xi (G_1))$ by $c_1(\Xi (G_1)) + rp_2^*c_1(P)$ . Thus, in order to find the desired $P$ , we have to show that

(13) \begin{align} c_1(G_2) - c_1(\Xi (G_1)) \in r \cdot p_2^*(H^2(X, \mathbb{Z}) \cap H^{1,1}(X)). \end{align}

The equality $p_2^*p_{2*}(\Xi (G_1) \otimes G_2^\vee ) \cong \Xi (G_1) \otimes G_2^\vee$ , which holds since $\Xi (G_1)$ is a $Y_2$ -sheaf, already implies that $r(c_1(G_2)-c_1(\Xi (G_1)))$ lies in $r \cdot p_2^*(H^2(X, \mathbb{Z}) \cap H^{1,1}(X))$ , which implies that

\begin{align*} c_1(G_2) - c_1(\Xi (G_1)) \in p_2^*(H^2(X, \mathbb {Z}) \cap H^{1,1}(X)). \end{align*}

To get the factor $r$ , we see that the image of $c_1(G_2) - c_1(\Xi (G_1))$ in $H^2(Y_2, \mu _r)$ is $p_2^*w - p_2^*w=0$ , by the assumption $w(Y_1) = w(Y_2) = w$ and [Reference YoshiokaYos06, Lemma 1.3, 1.8]. Using the explicit form of $H^2(Y_2, \mathbb{Z}) = p_2^*H^0(X,\mathbb{Z}) \oplus p_2^*H^2(X, \mathbb{Z})$ , we see that (13) is satisfied. This shows that we can pick $\Xi$ such that it preserves the Chern character. For the rest of the proof, we will use this $\Xi$ .

It is now not hard to show that

\begin{align*} \frac {\chi (X,p_{1*}(F \otimes G_1)(mH))}{{\mathrm {rk}}(F)r} = \frac {\chi (X,p_{2*}(\Xi (F) \otimes G_2)(mH))}{{\mathrm {rk}}(F) r} + c, \end{align*}

where $c$ is a constant that does not depend on $F$ . Therefore $\Xi$ preserves stability. Since $\Xi$ preserves the Chern character and the stability condition, we obtain an isomorphism

\begin{align*} \phi : M_{Y_1,\xi /r}^H(r,\xi, c_2) \cong M_{Y_2,\xi /r}^H(r,\xi, c_2) \end{align*}

for any lift $\xi$ of $w$ and any $c_2 \in \mathbb{Z}$ .

Suppose there exists a universal sheaf $\mathcal{E}_1$ on $Y_1 \times M_1$ , where $M_i:=M_{Y_i,\xi /r}^H(r,\xi, c_2)$ . Then

\begin{align*} \mathcal {E}_2 := \phi _* \Xi (\mathcal {E}_1) \end{align*}

is a universal sheaf on $Y_2 \times M_2$ , where we denote the base-changed versions of $\phi, \Xi$ by the same symbol. By the second identity of Lemma 4.8, we have

\begin{align*} Rp_{2*} R{\mathcal {H}\textit {om}}(\Xi (\mathcal {E}_1),\Xi (\mathcal {E}_1)) \cong Rp_{1*} R{\mathcal {H}\textit {om}}(\mathcal {E}_1, \mathcal {E}_1). \end{align*}

This readily implies $\phi ^* T_{M_2}^{{\rm vir}} \cong T_{M_1}^{{\rm vir}}$ . Note that this reasoning does not require the universal sheaf $\mathcal{E}_1$ to exist globally on $Y_1 \times M_1$ . This equality, and Siebert’s formula [Reference SiebertSie04, Theorem 4.6], imply that the virtual classes defined by both moduli problems agree,Footnote ¹⁸

(14) \begin{align} \phi ^* [M_{Y_2,\xi /r}^H(r,\xi, c_2)]^{{\rm vir}} = [M_{Y_1,\xi /r}^H(r,\xi, c_2)]^{{\rm vir}}. \end{align}

Note that it is not necessary to establish that

commutes, and we will therefore not do this.

To complete this part of the proof, we should check that the invariants themselves agree. Lemma 4.8 implies that

\begin{align*} \pi _2^*\phi ^*(\mathcal{E}_2^{\otimes r} \otimes \det (\mathcal{E}_2)^{-1}) & \cong (\pi _2^* \phi ^*\mathcal{E}_2)^{\otimes r} \otimes \det (\pi _2^* \phi ^*\mathcal{E}_2)^{-1})\\ & \cong (\pi _2^*\Xi (\mathcal{E}_1))^{\otimes r} \otimes \det (\pi _2^*\Xi (\mathcal{E}_1))^{-1}\\ & \cong (\pi _1^*\mathcal{E}_1 \otimes L)^{\otimes r} \otimes \det (\pi _1^*\mathcal{E}_1 \otimes L)^{-1}\\ & \cong \pi _1^*( \mathcal{E}_1^{\otimes r} \otimes \det (\mathcal{E}_1)^{-1}). \end{align*}

Note that it is crucial that we work with the ‘normalized’ universal sheaves. This gives us that, for any $\alpha \in H^*(X, \mathbb{Q})$ and $k \in \mathbb{Z}$ , we have

\begin{align*} \pi _{M_1*} \Big ( \pi _X^* \alpha \cap \mathrm {ch}_k(p_{1*} (\mathcal {E}_1 \otimes \det (\mathcal {E}_1)^{-\frac {1}{r}} )) \Big ) = \phi ^* \pi _{M_2*} \Big ( \pi _X^* \alpha \cap \mathrm {ch}_k(p_{2*} (\mathcal {E}_2 \otimes \det (\mathcal {E}_2)^{-\frac {1}{r}} )) \Big ), \end{align*}

where $\pi _X : M_i \times X \to X$ is the projection. We conclude that, for any $\boldsymbol{a}$ , $\boldsymbol{\alpha }$ as in Definition4.6, we have

\begin{align*} \int _{[M_{Y_1,\xi /r}^H(r,\xi, c_2)]^{{\rm vir}}} \mathsf{P}_{{\rm vd}(r,\xi, c_2),\boldsymbol{a}}(r,X,\boldsymbol{\alpha };M_{Y_1,\xi /r}^H(r,\xi, c_2),\mathcal{E}_1) \\ = \int _{[M_{Y_2,\xi /r}^H(r,\xi, c_2)]^{{\rm vir}}} \mathsf{P}_{{\rm vd}(r,\xi, c_2),\boldsymbol{a}}(r,X,\boldsymbol{\alpha };M_{Y_2,\xi /r}^H(r,\xi, c_2),\mathcal{E}_2). \end{align*}

The second part of the proposition follows [Reference Jiang and KoolJK21, Proposition 3.1]. Let $\pi : Y \rightarrow X$ be a degree $r$ Brauer–Severi variety with $w(Y)=w$ and let $\xi, \xi ^{\prime} \in H^2(X,\mathbb{Z})$ be two representatives of $w$ (Lemma 3.1). Then $\xi ^{\prime} = \xi +r\gamma$ for some $\gamma \in H^2(X,\mathbb{Z})$ . For any $Y$ -sheaf $F$ we have

\begin{align*} e^{\frac {\xi ^{\prime}}{r}} {\mathrm {ch}}_G(F) = (r,\xi ^{\prime},\tfrac {1}{2} \xi ^{\prime 2} - c_2^{\prime}) \end{align*}

if and only if

\begin{align*} e^{\frac {\xi }{r}} {\mathrm {ch}}_G(F) = (r,\xi, \tfrac {1}{2} \xi ^{2} - c_2), \quad c_2^{\prime} =c_2 + (r-1) \gamma \xi +\tfrac {1}{2} r (r-1) \gamma^2. \end{align*}

In particular, $M_{Y,\xi ^{\prime}/r}^H(r,\xi ^{\prime},c_2^{\prime}) = M_{Y,\xi /r}^H(r,\xi, c_2)$ and ${\rm vd}(r,\xi ^{\prime},c_2^{\prime}) = {\rm vd}(r,\xi, c_2)$ . Summing over all $c_2$ , $c_2^{\prime}$ , the result follows.

The independence of choice of $Y$ is perhaps not surprising, as this is just a particular model for the theory of twisted sheaves that we are using (although it is non-trivial that the Chern classes behave well under the equivalence discussed in Proposition 4.9). The independence of the choice of $\xi$ is also interesting: here we really need that we are dealing with generating functions and sum over all $c_2$ . A different choice leads to a permutation of the terms of the generating function.

We make some observations on the generating functions in Definitions4.5 and 4.6.

• • The ${\rm SL}_r$ generating function is invariant under replacing $\mathcal{E}$ by $\mathcal{E} \otimes \mathcal{L}$ for any line bundle $\mathcal{L}$ on $X \times M$ . Let $L$ be a line bundle on $X$ and suppose $- \otimes L$ preserves stability, i.e. it induces an isomorphism

  \begin{align*} \bigsqcup _{c_2} M_{X}^H(r,c_1,c_2) \to \bigsqcup _{c_2} M_{X}^H(r,c_1+rc_1(L),c_2). \end{align*}
  This happens, for example, when (i) $L$ is a multiple of $H$ , or (ii) when $\gcd (r,c_1H)=1$ so that Gieseker stability and $\mu$ -stability coincide [Reference Huybrechts and LehnHL10, Lemma 1.2.13, 1.2.14]. Then
  \begin{align*} \mathsf {Z}_{(X,H),c_1}^{\mathrm {SL}_r, \mathsf {P}}(q) = \mathsf {Z}_{(X,H),c_1+rc_1(L)}^{\mathrm {SL}_r, \mathsf {P}}(q). \end{align*}

• • Suppose $\alpha (Y) \in {\rm Br}(X)$ has order $r\gt 1$ . Then stability is automatic by Remark 3.5 and we write

  \begin{align*} \mathsf {Z}_{X,w}^{\mathrm {PGL}_r, \mathsf {P}}(q) = \mathsf {Z}_{(X,H),w}^{\mathrm {PGL}_r, \mathsf {P}}(q). \end{align*}

• • Many interesting virtual invariants can be obtained from appropriate choices of formal insertions $\mathsf{P}$ by the virtual Hirzebruch–Riemann–Roch theorem [Reference Ciocan-Fontanine and KapranovCFK09, Reference Fantechi and GöttscheFG10]:

  1. – virtual Euler characteristic $e^{{\rm vir}}$ , Hirzebruch genus $\chi _{-y}^{{\rm vir}}$ , elliptic genus $Ell^{{\rm vir}}$ . These were defined by Fantechi and Göttsche [Reference Fantechi and GöttscheFG10] and studied in the ${\rm SL}_r$ case in, e.g. [Reference Göttsche and KoolGK18, Reference Göttsche and KoolGK20b, Reference Göttsche, Kool and LaarakkerGKL24];

  2. – Donaldson invariants and $K$ -theoretic Donaldson invariants studied in algebraic geometry by Göttsche, Nakajima and Yoshioka in the ${\rm SL}_r$ case in [Reference Göttsche, Nakajima and YoshiokaGNY11, Reference Göttsche, Nakajima and YoshiokaGNY08, Reference Göttsche, Nakajima and YoshiokaGNY09];

  3. – virtual cobordism class defined by Shen [Reference ShenShe16] and studied in the ${\rm SL}_r$ case in [Reference Göttsche and KoolGK19];

  4. – virtual Segre and Verlinde numbers [Reference Göttsche and KoolGK22, Reference Göttsche, Kool and WilliamsGKW21];

  5. – Virasoro operators [Reference Bojko, Lim and MoreiraBLM24, Reference van BreevBr23].

We now make a comparison between moduli of sheaves on a trivial Brauer–Severi variety $\mathbb{P}(E^\vee ) \to X$ and moduli of sheaves on $X$ .

Proposition 4.10. Let $(X,H)$ be a smooth polarized surface satisfying $H_1(X,\mathbb{Z}) = 0$ and consider a projective bundle $\pi : Y = \mathbb{P}(E^\vee ) \to X$ for a rank $r$ vector bundle $E$ on $X$ . Let $w := w(Y) \in H^2(X,\mu _r)$ and let $c_1 \in H^2(X,\mathbb{Z})$ be a (necessarily algebraic) class representing $w$ . Suppose $\gcd (r,c_1 H)=1$ . Then there exists an isomorphism between Yoshioka and Gieseker–Maruyama–Simpson moduli spaces

\begin{align*} \phi : M_{Y,c_1/r}^H(r,c_1,c_2) \cong M_{X}^H(r,c_1,c_2). \end{align*}

Moreover, under this isomorphism we have the following comparison of virtual fundamental classes and (normalized) universal sheaves: Footnote ¹⁹

\begin{align*} \phi ^* T_{M_{X}^H(r,c_1,c_2)}^{{\rm vir}} \cong T_{M_{Y,c_1/r}^H(r,c_1,c_2)}^{{\rm vir}}, \quad \phi ^* [M_{X}^H(r,c_1,c_2)]^{{\rm vir}} & = [M_{Y,c_1/r}^H(r,c_1,c_2)]^{{\rm vir}}, \\ \pi ^* \phi ^* \Big ( (\mathcal{E}_2)^{\otimes r} \otimes \det (\mathcal{E}_2)^{-1} \Big ) \cong \mathcal{E}_1^{\otimes r} \otimes \det (\mathcal{E}_1)^{-1}, \end{align*}

where $\pi$ and $\phi$ denote the appropriately base-changed morphisms.

Proof. There exists an equivalence of categories [Reference YoshiokaYos06, Lemma 1.7]

\begin{align*} \Xi : \mathrm {Coh}(X,Y) \to \mathrm {Coh}(X), \quad F \mapsto \pi _*(F(-1)). \end{align*}

We fix the $B$ -field $\xi :=c_1$ . Using $G = \pi ^*E(1)$ and the projection formula, a direct calculation shows that for any $Y$ -sheaf $F$ , we have

\begin{align*} \mathrm {ch}_G^\xi (F) = \mathrm {ch}(\Xi (F)). \end{align*}

Hence ${{\rm rk}}(F) ={{\rm rk}}(\Xi (F))$ . Suppose this rank is positive. Then, using the Hirzebruch–Riemann–Roch theorem, we find that the reduced Hilbert polynomials are related by

\begin{align*} {\chi (\pi _*(F \otimes G^\vee )(mH)) \over a_2^G(F)} = & {\chi (\Xi (F)(mH)) \over a_2(\Xi (F))} + {c_1(E^\vee )H \over r H^2} m \\ & + {c_1(E^\vee )c_1(X) \over 2rH^2} + {{\rm ch_2}(E^\vee ) \over r H^2} + {c_1(E^\vee )c_1(\Xi (F)) \over r{{\rm rk}}(\Xi (F)) H^2}. \end{align*}

We would like to prove the following claim: for any torsion free sheaf $F$ on $X$ , we have

\begin{align*} \frac {\chi (F^{\prime}(mH))}{a_2(F^{\prime})} + \frac {1}{r H^2} \frac {c_1(E^\vee )c_1(F^{\prime})}{{\mathrm {rk}}(F^{\prime})} \lt \frac {\chi (F(mH))}{a_2(F)} + \frac {1}{r H^2} \frac {c_1(E^\vee )c_1(F)}{{\mathrm {rk}}(F)}, \end{align*}

for any subsheaf $0 \neq F^{\prime} \subsetneq F$ , if and only if $F$ is $\mu$ -stable [Reference Huybrechts and LehnHL10, Definition 1.2.12]. We may replace $\mathbb{P}(E^\vee )$ by $\mathbb{P}(E^\vee \otimes \mathcal{L}^N)$ , where $\mathcal{L}$ is an ample line bundle on $X$ and $N \in \mathbb{Z}_{\gt 0}$ . This replaces $c_1(E^\vee )$ by $N r c_1(\mathcal{L}) + c_1(E^\vee )$ which is ample for $N \gg 0$ . In other words, we may assume $E$ is chosen such that $c_1(E^\vee )$ is ample. In particular, for any torsion free sheaf $F$ on $X$ and subsheaf $F^{\prime}$ with ${{\rm rk}}(F^{\prime}) ={{\rm rk}}(F)$ , we have

\begin{align*} c_1(E^\vee )(c_1(F) - c_1(F^{\prime})) \geq 0, \end{align*}

by the Nakai–Moishezon criterion. Assuming $\gcd ({{\rm rk}}(F),c_1(F) H)=1$ , the claim then easily follows. We deduce that $\Xi$ induces an isomorphism of moduli spacesFootnote ²⁰

\begin{align*} \phi : M_{Y,c_1/r}^H(r,c_1,c_2) \stackrel {\cong }{\to } M_{X}^H(r,c_1,c_2). \end{align*}

The second part of the proposition is established exactly as in the proof of Proposition 4.9.