Fourier-Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes

TL;DR

This paper explores how vector bundles with Hitchin pair structures on curves or surfaces induce related structures on punctual Hilbert schemes, extending stability and reconstruction results to these tautological Hitchin pairs.

Contribution

It introduces a method to induce Hitchin pair structures on tautological bundles over Hilbert schemes, generalizing stability and reconstruction results.

Findings

Tautological bundles inherit Hitchin pair structures.

Higgs bundles induce logarithmic Higgs bundles on Hilbert schemes.

Stability and reconstruction results extend to tautological Hitchin pairs.

Abstract

Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ of $V$. In particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the Hilbert scheme $X^{[n]}$. We then show that the known results on stability of tautological bundles and reconstruction from tautological bundles generalize to tautological Hitchin pairs.

Equations

$$E^{[n]}\otimes T_{X^{[n]}}(-\log B_n)\cong E^{[n]}\otimes T_X^{[n]}\stackrel{\nu }{\to }(E\otimes T_X{)}^{[n]}\stackrel{{\vartheta }^{[n]}}{\to }{E}^{[n]}$$

$$(E,\vartheta {)}^{[n]}\cong (F,\eta {)}^{[n]}⟹(E,\vartheta )\cong (F,\eta ).$$

$$\bigl{(}\vartheta\circ(\vartheta\otimes\operatorname{\mathsf{id}}_{V})\bigr{)}_{\mid\bigwedge\nolimits^{2}V}\colon E\otimes\bigwedge\nolimits^{2}V\hookrightarrow E\otimes V\otimes V\xrightarrow{\vartheta\otimes\operatorname{\mathsf{id}}_{V}}E\otimes V\xrightarrow{\vartheta}E\,,$$

$$E\otimes V\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{For}:{\mathrm{Hi}}^V(X)\to \mathrm{Coh}(X)$$

$$E\stackrel{\zeta }{\to }E\otimes V\stackrel{\zeta \otimes {\mathrm{id}}_V}{\to }E\otimes V\otimes V\twoheadrightarrow E\otimes {\bigwedge }^{2}V$$

$$\displaystyle\operatorname{\mathsf{Hi}}^{V}(X)\to\operatorname{\mathsf{Hi}}_{V^{\vee}}(X)\,,\quad(E,\vartheta)\mapsto(E,\vartheta^{\prime})\,,\quad\vartheta^{\prime}:=\bigl{(}E\xrightarrow{\operatorname{\mathsf{id}}_{E}\otimes\eta_{V}}E\otimes V\otimes V^{\vee}\xrightarrow{\vartheta\otimes\operatorname{\mathsf{id}}_{V^{\vee}}}E\otimes V^{\vee}\bigr{)}\,,$$

$$\displaystyle\operatorname{\mathsf{Hi}}_{V^{\vee}}(X)\to\operatorname{\mathsf{Hi}}^{V}(X)\,,\quad(E,\zeta)\mapsto(E,\zeta^{\prime})\,,\quad\zeta^{\prime}:=\bigl{(}E\otimes V\xrightarrow{\zeta\otimes\operatorname{\mathsf{id}}_{V}}E\otimes V^{\vee}\otimes V\xrightarrow{\operatorname{\mathsf{id}}_{E}\otimes\varepsilon_{V}}E\bigr{)}\,,$$

$$\varphi^{\#}\colon\operatorname{\mathsf{Hi}}^{W}(X)\to\operatorname{\mathsf{Hi}}^{V}(X)\quad,\quad(E,\vartheta)\mapsto\bigl{(}E,\vartheta\circ(\operatorname{\mathsf{id}}_{E}\otimes\varphi)\bigr{)}\,.$$

$$\alpha :={\alpha }_{f,E,V}:f^{\ast }E\otimes f^{\ast }V\stackrel{\sim }{\to }{f}^{\ast }(E\otimes V).$$

$$(M{\otimes }_A{A}^{\prime }){\otimes }_{A^{\prime }}(N{\otimes }_A{A}^{\prime })\to (M{\otimes }_{A}N){\otimes }_A{A}^{\prime },(m\otimes a)\otimes (n\otimes b)\mapsto (m\otimes n)\otimes (ab).$$

$$f^{\ast }:{\mathrm{Hi}}^V(X)\to {\mathrm{Hi}}^{f^{\ast }V}(X^{\prime }),(E,\vartheta )\mapsto (f^{\ast }E,f^{\ast }\vartheta \circ \alpha ).$$

$$\nu :={\nu }_{\pi ,E,V}:{\pi }_{\ast }E\otimes {\pi }_{\ast }V\to {\pi }_{\ast }(E\otimes V).$$

$$\Gamma ({\pi }^{-1}U,E){\otimes }_A\Gamma ({\pi }^{-1}U,V)\stackrel{\cup }{\to }\Gamma ({\pi }^{-1}U,E\otimes V).$$

$$\check{\vartheta }:={\pi }_{\ast }\vartheta \circ \nu :{\pi }_{\ast }E\otimes {\pi }_{\ast }V\to {\pi }_{\ast }E.$$

$$\bigl{(}\check{\vartheta}\circ(\check{\vartheta}\otimes\operatorname{\mathsf{id}}_{\pi_{*}V})\bigr{)}_{\mid\pi_{*}E\otimes\bigwedge\nolimits^{2}(\pi_{*}V)}=0\,.$$

$${\pi }_{\ast }(E\otimes V)\otimes {\pi }_{\ast }V\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\check{\vartheta }\circ (\check{\vartheta }\otimes {\mathrm{id}}_V)$$

$$=({\pi }_{\ast }\vartheta )\circ ({\pi }_{\ast }(\vartheta \otimes {\mathrm{id}}_V))\circ {\nu }_{E\otimes V,V}\circ ({\nu }_{E,V}\otimes {\mathrm{id}}_{{\pi }_{\ast }V})$$

$$=({\pi }_{\ast }(\vartheta \circ (\vartheta \otimes {\mathrm{id}}_V)))\circ {\nu }_{E,V\otimes V}\circ ({\mathrm{id}}_{{\pi }_{\ast }E}\otimes {\nu }_{V,V}).$$

$${\nu }_{E,V\otimes V}({\mathrm{id}}_{{\pi }_{\ast }E}\otimes {\nu }_{V,V})({\pi }_{\ast }E\otimes {\bigwedge }^2({\pi }_{\ast }V))\subset {\pi }_{\ast }(E\otimes {\bigwedge }^{2}V).$$

$$({\pi }_{\ast }(\vartheta \circ (\vartheta \otimes {\mathrm{id}}_V)){)}_{\mid {\pi }_{\ast }(E\otimes {\bigwedge }^{2}V)}=0.$$

$${\pi }_{\ast }:{\mathrm{Hi}}^V(X)\to {\mathrm{Hi}}^{{\pi }_{\ast }V}(Y),{\pi }_{\ast }(E,\vartheta )=({\pi }_{\ast }E,\check{\vartheta }).$$

$$X^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\beta :={\beta }_E:f^{\ast }{\pi }_{\ast }E\to {\pi }_{\ast }^{\prime }{f}^{\prime \ast }E.$$

$$B^{\prime }$$

$$(M_A){\otimes }_A{A}^{\prime }\to (M{\otimes }_B{B}^{\prime }{)}_{A^{\prime }},m\otimes a\mapsto m\otimes {\phi }^{\prime }(a).$$

$${\beta }_E:f^{\ast }{\pi }_{\ast }(E,\vartheta )\stackrel{\sim }{\to }{\beta }_V^{\#}{\pi }_{\ast }^{\prime }{f}^{\prime \ast }(E,\vartheta )$$

$$f^{\ast }{\pi }_{\ast }E\otimes f^{\ast }{\pi }_{\ast }V\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(M_A{\otimes }_A{A}^{\prime }){\otimes }_{A^{\prime }}(N_A{\otimes }_A{A}^{\prime })\to (M{\otimes }_B{B}^{\prime }{)}_{A^{\prime }},(m\otimes a_1)\otimes (n\otimes a_2)\mapsto \vartheta (m\otimes n)\otimes {\phi }^{\prime }(a_1{a}_2).\text{qed}$$

Full text

1. Introduction

Let $X$ be a smooth projective variety with $\operatorname{\mathsf{dim}}X\leq 2$. In this case, the Hilbert scheme $X^{[n]}$ of $n$ points on $X$ is a smooth projective variety of dimension $\operatorname{\mathsf{dim}}X^{[n]}=n\cdot\operatorname{\mathsf{dim}}X$. If $X$ is a curve, $X^{[n]}$ coincides with the symmetric product $X^{(n)}:=X^{n}/\mathfrak{S}_{n}$, where $\mathfrak{S}_{n}$ is the group of permutations of $\{1,\dots,n\}$. If $X$ is a surface, then $X^{[n]}$ is a resolution of the singularities of $X^{(n)}$ via the Hilbert–Chow morphism $\mu\colon X^{[n]}\to X^{(n)}$. The points parametrizing non-reduced sub-schemes of $X$ form a divisor $B_{n}\subset X^{[n]}$. Given a vector bundle $E$ on $X$, we get a vector bundle $E^{[n]}$ on $X^{[n]}$ which is the direct image of the pullback of $E$ to the universal subscheme $\Xi_{n}\subset X\times X^{[n]}$. These vector bundles on $X^{[n]}$ play a crucial role in the investigation of the topology and geometry of the Hilbert schemes [Leh99, LS01, LS03], but are also useful tools for studying properties of the bundles on $X$ itself; see, for example [Voi02, EL15, Ago17].

Higgs bundles constitute an extensively studied topic; they appear in algebraic geometry, differential geometry, symplectic geometry and also in representation theory [Ngo10]. We recall that the Higgs bundles on Riemann surfaces were introduced by Hitchin [Hi87] and the Higgs bundles on higher dimensional complex manifolds were introduced by Simpson [Si88].

It turns out that a Higgs field $E\to E\otimes\Omega_{X}$ of a vector bundle $E$ on $X$ does not, in general, induce a Higgs field of $E^{[n]}$; see Subsection 3.3. However we observe that it induces a logarithmic Higgs field on $E^{[n]}$, that is a homomorphism $E^{[n]}\to E^{[n]}\otimes\Omega_{X^{[n]}}(\operatorname{\mathsf{log}}B_{n})$ or, equivalently, a homomorphism $E^{[n]}\otimes\operatorname{\mathsf{T}\!}_{X^{[n]}}(-\operatorname{\mathsf{log}}B_{n})\to E^{[n]}$.

The construction is as follows. As proved by Stapleton [Sta16, Thm. B], there is an isomorphism $\operatorname{\mathsf{T}\!}_{X^{[n]}}(-\operatorname{\mathsf{log}}B_{n})\cong(\operatorname{\mathsf{T}\!}_{X})^{[n]}$, identifying the sheaf of logarithmic vector fields on $X^{[n]}$ with the tautological bundle associated to the tangent sheaf of $X$. Now, the logarithmic Higgs field on $E^{[n]}$ associated to a Higgs field $\vartheta\colon E\otimes\operatorname{\mathsf{T}\!}_{X}\to E$ is given by the composition

[TABLE]

where $\nu\colon E^{[n]}\otimes\operatorname{\mathsf{T}\!}_{X}^{[n]}\to(E\otimes\operatorname{\mathsf{T}\!}_{X})^{[n]}$ is the cup product relative to the projection morphism $\operatorname{\mathsf{pr}}_{X^{[}n]}\colon\Xi_{n}\to X^{[n]}$; compare Subsection 2.7 and Subsection 3.2. We denote this induced Higgs field by $\vartheta^{\{n\}}\colon E^{[n]}\otimes\operatorname{\mathsf{T}\!}_{X^{[n]}}(-\operatorname{\mathsf{log}}B_{n})\to E^{[n]}$, and get the tautological logarithmic Higgs bundle $(E,\vartheta)^{[n]}:=(E,\vartheta)$.

There is the more general notion of a $V$-cotwisted Hitchin pair, for $V$ a vector bundle of $X$, such that a Higgs bundle on $X$ is exactly a $\operatorname{\mathsf{T}\!}_{X}$-cotwisted Hitchin pair, and a logarithmic Higgs bundle on $X^{[n]}$ is exactly a $\operatorname{\mathsf{T}\!}_{X^{[n]}}(-\operatorname{\mathsf{log}}B_{n})$-cotwisted Hitchin pair; see Subsection 2.1. The above construction generalizes in such a way that for a $V$-cotwisted Hitchin pair $(E,\vartheta)$ on $X$, we get a tautological $V^{[n]}$-cotwisted Hitchin pair $(E,\vartheta)^{[n]}=(E^{[n]},\vartheta^{\{n\}})$ on $X^{[n]}$; see Subsection 3.2.

In this article, we proof that basically all the known results on stability of tautological bundles and reconstruction from tautological bundles lift to results for tautological Hitchin pairs. As probably the most interesting special case, we get results on tautological logarithmic Higgs bundles. Concretely, we proof the following results.

Theorem 1.1**.**

Let $C$ be a smooth projective curve of genus $\operatorname{\mathsf{g}}(C)\geq 2$, and let $(E,\vartheta)$, $(F,\eta)$ be semi-stable $V$-cotwisted Hitchin bundles for some vector bundle $V\in\operatorname{\mathsf{VB}}(C)$. Then, for every $n\in\mathbb{N}$, we have

[TABLE]

For a scheme $Y$, by $\operatorname{\mathsf{VB}}(Y)$ and $\operatorname{\mathsf{Coh}}(Y)$ we denote the category of vector bundles and coherent sheaves respectively on $Y$.

Theorem 1.2**.**

Let $C$ be an elliptic curve, and let $(E,\vartheta)$, $(F,\eta)$ be $V$-cotwisted Hitchin bundles for some vector bundle $V\in\operatorname{\mathsf{VB}}(C)$. Then, for every $n\in\mathbb{N}$, we have

[TABLE]

Theorem 1.3**.**

Let $X$ be a smooth quasi-projective variety of dimension $\operatorname{\mathsf{dim}}X\geq 2$, let $V\in\operatorname{\mathsf{VB}}(X)$, and let $(E,\vartheta),(F,\eta)$ be $V$-cotwisted Hitchin pairs for some vector bundle $V\in\operatorname{\mathsf{VB}}(C)$, such that the coherent sheaves $E$ and $F$ are reflexive. Then, for every $n\in\mathbb{N}$,

[TABLE]

Let $C$ be a smooth projective curve. In the group $\operatorname{\mathsf{N}}^{1}(C^{(n)})$ of divisors modulo numerical equivalence, we consider the ample class $H_{n}$ that is represented by $C^{(n-1)}+x\subset C^{(n)}$ for any $x\in C$; see [Kru18b, Sect. 1.3] for details.

Theorem 1.4**.**

Let $C$ be a smooth projective curve, and let $(E,\vartheta)$ be a Hitchin bundle on $C$. Then for every $n\in\mathbb{N}$, the following two statements hold.

1. (i)

If $(E,\vartheta)$ is stable and $\mu(E)\notin[-1,n-1]$, then the logarithmic Higgs bundle $(E^{[n]},\vartheta^{\{n\}})$ is stable with respect to $H_{n}$. 2. (ii)

If $(E,\vartheta)$ is semi-stable and $\mu(E)\notin(-1,n-1)$, then the logarithmic Higgs bundle $(E^{[n]},\vartheta^{\{n\}})$ is semi-stable with respect to $H_{n}$.

The paper is organized as follows. In Section 2, we collect basic notions and results on Hitchin pairs needed for later use. In particular, we define the push-forward of Hitchin pairs along flat and finite morphisms, and define Fourier–Mukai transforms of Hitchin pairs for a certain class of kernels; see Subsection 2.7 and Subsection 2.8. In Section 3, we explain, in some more detail than done in this introduction, how to get induced structures of Hitchin pairs and logarithmic Higgs bundles on tautological bundles on Hilbert schemes of points. All four of our results listed above have already been proved in the special case of ordinary sheaves without the structure of a Hitchin pair. (A coherent sheaf is the same as a [math]-cotwisted Hitchin pair.) For Theorem 1.1, see [BN12] and [BN17, Sect. 2]; for Theorem 1.2, see [Kru18a, Thm. 1.5]; for Theorem 1.3, see [Kru18a, Thm. 3.2 & Rem. 3.5]; for Theorem 1.4, see [Kru18b]. In all cases, the proofs can be amended in such a way that they work for Hitchin pairs. We follow different approaches in explaining how to amend the proofs. For Theorem 1.1 and Theorem 1.2, we give full proofs in Section 4. The reader should be able to follow these proofs without going back to [BN12], [BN17] or [Kru18a], though some steps of the arguments might be carried out in greater details in these articles. In contrast, for the proofs of Theorem 1.3 and Theorem 1.4, we only explain the extra ingredients needed to lift the proofs from the cases of ordinary sheaves to Hitchin pairs, and where to insert these ingredients. Hence, for following these proofs, the reader should at the same time have a look at the relevant parts of [Kru18a] and [Kru18b].

In the final Subsection 6.2, we remark that also a result of Stapleton [Sta16] on stability of tautological bundles on Hilbert schemes of points on surfaces extends to Hitchin bundles.

2. Preliminaries on Hitchin pairs

Throughout this section, let $X$ be a scheme over the complex numbers $\mathbb{C}$. Usually, we will work with varieties, but at one point in Subsection 4.1 we will have to consider Hitchin pairs on infinitesimal neighborhoods of a diagonal, which is why we choose the greater generality here.

2.1. Basic Definitions

Let $V\in\operatorname{\mathsf{VB}}(X)$ be a vector bundle. A $V$-cotwisted Hitchin pair on $X$ is a pair $(E,\vartheta)$ consisting of a coherent sheaf $E\in\operatorname{\mathsf{Coh}}(X)$ and an $\mathcal{O}_{X}$-linear map $\vartheta\colon E\otimes V\to V$ such that the composition

[TABLE]

where $\bigwedge^{2}V\subset V\otimes V$ is the second exterior product, vanishes. A Hitchin pair $(E,\vartheta)$ where $E$ is a vector bundle is also called a Hitchin bundle.

A morphism between two $V$-cotwisted Hitchin pairs $(E,\vartheta)$ and $(F,\eta)$ on $X$ is an $\mathcal{O}_{X}$-linear map $\gamma\colon E\to F$ such that the following diagram commutes

[TABLE]

We denote the category of $V$-cotwisted Hitchin pairs on $X$ by $\operatorname{\mathsf{Hi}}^{V}(X)$. It is an abelian category with kernels and cokernels given by the kernels and cokernels of the underlying morphisms of coherent sheaves, equipped with the induced $\mathcal{O}_{X}$-linear homomorphism. In other words, the forgetful functor

[TABLE]

is exact.

Remark \theremark.

In the literature, usually $V$-twisted Hitchin pairs are considered instead of $V$-cotwisted Hitchin pairs. A $V$-twisted Hitchin pair is a pair $(E,\zeta)$ consisting of $E\in\operatorname{\mathsf{Coh}}(X)$ and an $\mathcal{O}_{X}$-linear morphism $\zeta\colon E\to E\otimes V$ such that the composition

[TABLE]

vanishes. However, the category $\operatorname{\mathsf{Hi}}_{V^{\vee}}(X)$ of $V^{\vee}$-twisted Hitchin pairs is equivalent to $\operatorname{\mathsf{Hi}}^{V}(X)$ via the pair of mutually inverse exact functors

[TABLE]

where $\eta_{V}\colon\mathcal{O}_{X}\to V\otimes V^{\vee}$ sends any $f$ to $f\cdot\operatorname{\mathsf{id}}_{V}$, and $\varepsilon_{V}\colon V^{\vee}\otimes V\to\mathcal{O}_{X}$ is the trace map. The reason that we prefer to work with $V$-cotwisted sheaves is that they allow an easier formulation of a push-forward functor (under certain circumstances); see Subsection 2.7.

2.2. Hitchin subsheaves

Let $(E,\vartheta)$ be a $V$-cotwisted Hitchin pair on $X$. A coherent subsheaf $A\subset E$ is called a Hitchin subsheaf if $\vartheta(A\otimes V)\subset A$. A Hitchin subsheaf carries itself the structure of a $V$-cotwisted Hitchin pair $(A,\vartheta_{\mid A})$ which makes the inclusion $A\hookrightarrow E$ a morphism of Hitchin pairs. Furthermore, the quotient $E/A$ carries the structure of a Hitchin pair $(E/A,\bar{\vartheta})$ such that the quotient map $E\twoheadrightarrow E/A$ is a morphism of Hitchin pairs.

For later use, we note the following quite obvious statements.

Lemma \thelemma.

Let $(E,\vartheta)$ be a Hitchin pair, and let $A\subset E$ be a Hitchin subsheaf.

1. (i)

If $B\subset E$ is another Hitchin subsheaf, then the intersection $A\cap B\subset E$ is a Hitchin subsheaf too. 2. (ii)

Let $(F,\eta)$ be another Hitchin pair, and let $\varphi\colon E\to F$ be a morphism of Hitchin pairs. Then $\varphi(A)\subset F$ is a Hitchin subsheaf.

2.3. Higgs bundles

Let $X$ be a smooth variety with tangent bundle $\operatorname{\mathsf{T}\!}_{X}$. A Higgs bundle on $X$ is a $\operatorname{\mathsf{T}\!}_{X}$-cotwisted Hitchin pair $(E,\vartheta)$.

Given a reduced divisor $D\subset X$, we consider the sheaf of logarithmic vector fields $\operatorname{\mathsf{T}\!}_{X}(-\operatorname{\mathsf{log}}D)$. It is defined as the sheaf of vector fields which along the regular locus $D_{\mathsf{reg}}$ of $D$ are tangent to $D$. A logarithmic Higgs bundle (with respect to the divisor $D$) is a $\operatorname{\mathsf{T}\!}_{X}(-\operatorname{\mathsf{log}}D)$-cotwisted Hitchin pair $(E,\vartheta)$.

Logarithmic Higgs bundles on curves were first considered by Bottacin [Bo95]. He proved that a moduli space of logarithmic Higgs bundle on a curve has a natural Poisson structure.

We note that a logarithmic Higgs bundle is a parabolic Higgs bundle with trivial quasi-parabolic structure. Mochizuki has proved many important results on parabolic Higgs bundles [Mo06], [Mo14].

2.4. Change of the cotwisting bundle

Let $\varphi\colon V\to W$ be a morphism of vector bundles on $X$. There is an induced exact functor

[TABLE]

Lemma \thelemma.

Let $\varphi\colon V\to Q$ be a surjective morphism of vector bundles with kernel $K:=\operatorname{\mathsf{ker}}(p)$.

1. (i)

The functor $\varphi^{\#}\colon\operatorname{\mathsf{Hi}}^{Q}(X)\to\operatorname{\mathsf{Hi}}^{V}(X)$ is fully faithful. 2. (ii)

A Hitchin pair $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$ is in the essential image of the functor $\varphi^{\#}$ if and only if $\vartheta(E\otimes K)=0$. 3. (iii)

Let $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{Q}(X)$. A subsheaf $A\subset E$ is a Hitchin subsheaf of $(E,\vartheta)$ if and only if it is a Hitchin subsheaf of $\varphi^{\#}(E,\vartheta)$.

Proof.

Let $(E,\vartheta),(F,\eta)\in\operatorname{\mathsf{Hi}}^{Q}(X)$. By the surjectivity of $\varphi$, a $\mathcal{O}_{X}$-linear morphism $\gamma\colon E\to F$ is a morphism between the $Q$-cotwisted Hitchin pairs $(E,\vartheta)$ and $(F,\eta)$ if and only if it is a morphism between the $V$-cotwisted Hitchin pairs $\varphi^{\#}(E,\vartheta)$ and $\varphi^{\#}(F,\eta)$, which proves (i). Part (ii) is straight-forward to prove. Part (iii) follows directly from the surjectivity of $\varphi$ and the definition of Hitchin subsheaves. ∎

2.5. Pull-back of Hitchin pairs

Let $f\colon X^{\prime}\to X$ be a morphism, and let $E$ and $V$ be coherent sheaves on $X$. There is a natural isomorphism

[TABLE]

Concretely, if $f\colon\operatorname{\mathsf{Spec}}A^{\prime}\to\operatorname{\mathsf{Spec}}A$ is a morphism of affine schemes, so that $E=\widetilde{M}$, $V=\widetilde{N}$ for some $A$-modules $M$ and $N$, then $\alpha$ is given by the map

[TABLE]

Using the isomorphism $\alpha$, we can define the pull-back along $f$ on the level of Hitchin pairs as

[TABLE]

We often omit the isomorphism $\alpha$ in our notation and simply write $f^{*}\vartheta\colon f^{*}E\otimes f^{*}V\to f^{*}E$.

2.6. Hitchin pairs under tensor products

Let $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$ be a Hitchin pair, and let $P\in\operatorname{\mathsf{Coh}}(X)$. We write $P\otimes(E,\vartheta)$ for the Hitchin pair $(P\otimes E,\operatorname{\mathsf{id}}_{P}\otimes\vartheta)$. Clearly, this defines a functor $P\otimes\underline{\,\,\,\,}\colon\operatorname{\mathsf{Hi}}^{V}(X)\to\operatorname{\mathsf{Hi}}^{V}(X)$.

2.7. Push-forwards of Hitchin pairs

Let now $\pi\colon X\to Y$ be a morphism such that $\pi_{*}V$ is again a vector bundle. Note that this is always the case if $\pi$ is flat and finite. In the following, for every $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$, we will naturally equip $\pi_{*}E$ with the structure of a $\pi_{*}V$-cotwisted Hitchin pair. There is a natural morphism

[TABLE]

Over an open affine subset $\operatorname{\mathsf{Spec}}A=U\subset X$, it is given by the $A$-linear cup product

[TABLE]

Lemma \thelemma.

Let $\pi\colon X\to Y$ be a morphism of schemes, let $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$, and set

[TABLE]

Then the following vanishing statement holds:

[TABLE]

Proof.

From the description (2) of $\nu$ over open affine subsets, we see that the diagram

[TABLE]

commutes. Hence, we can rewrite the composition $\check{\vartheta}\circ(\check{\vartheta}\otimes\operatorname{\mathsf{id}}_{V})$ as

[TABLE]

It follows from (2) that $\nu_{V,V}(\bigwedge^{2}(\pi_{*}V))\subset\pi_{*}(\bigwedge^{2}V)$, hence

[TABLE]

Since $(E,\vartheta)$ is a Hitchin pair, we have that $(\vartheta\circ(\vartheta\otimes\operatorname{\mathsf{id}}_{V}))_{\mid E\otimes\bigwedge^{2}V}=0$, hence

[TABLE]

The combination of (3), (4), and (5) gives the desired vanishing. ∎

Corollary \thecorollary.

Let $\pi\colon X\to Y$ be a morphism of schemes, and let $V\in\operatorname{\mathsf{VB}}(X)$ be a vector bundle with the property that $\pi_{*}V$ is again a vector bundle. Then there is a push-forward functor

[TABLE]

In particular, if $\pi\colon X\to Y$ is flat and finite, there exists for every vector bundle $V\in\operatorname{\mathsf{VB}}(X)$ a push-forward functor $\pi_{*}\colon\operatorname{\mathsf{Hi}}^{V}(X)\to\operatorname{\mathsf{Hi}}^{\pi_{*}V}(Y)$ .

Remark \theremark.

Subsection 2.7 is the reason that we are working with cotwisted instead of twisted Hitchin pairs; compare Subsection 2.1. Indeed, in terms of twisted Hitchin pairs, the push-forward is a functor $\operatorname{\mathsf{Hi}}_{V}(X)\to\operatorname{\mathsf{Hi}}_{(\pi_{*}(V^{\vee}))^{\vee}}(Y)$, which makes it a bit inconvenient to formulate things in terms of twisted Hitchin pairs whenever a push-forward occurs.

Given a Cartesian diagram of schemes

[TABLE]

and a coherent sheaf $E\in\operatorname{\mathsf{Coh}}(X)$, we denote the natural base change morphism by

[TABLE]

In the case that all the schemes involved are affine, which means that (6) is the spectrum of a diagram of commutative rings of the form

[TABLE]

with $E=\widetilde{M}$ and $V=\widetilde{N}$ for some $B$-modules $M$ and $N$, the map $\beta$ is given by

[TABLE]

Lemma \thelemma.

Let $\pi$ be flat and finite, and let $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$. Then

[TABLE]

is an isomorphisms of $f^{*}\pi_{*}V$-cotwisted Hitchin pairs on $Y^{\prime}$.

Proof.

By the assumptions on $\pi$, the map $\beta_{E}\colon f^{*}\pi_{*}E\to\pi^{\prime}_{*}f^{\prime*}E$ is an isomorphism of coherent sheaves. It remains to proof that it is a morphism of Hitchin pairs, which amounts to checking the commutativity of the diagram

[TABLE]

For this, we may assume that (6) is given by the spectrum of (7), $E=\widetilde{M}$, and $V=\widetilde{N}$. Then, using the concrete descriptions (1), (2), and (8) of the maps $\alpha$, $\nu$, and $\beta$, it can be checked that both paths of the above diagram are given by

[TABLE]

Later, we usually omit the equivalence $\beta^{\#}\colon\operatorname{\mathsf{Hi}}^{f^{*}\pi_{*}V}(Y^{\prime})\xrightarrow{\sim}\operatorname{\mathsf{Hi}}^{\pi_{*}^{\prime}f^{\prime*}V}(Y^{\prime})$ when applying Subsection 2.7, and simple write $f^{*}\pi_{*}(E,\vartheta)\cong\pi^{\prime}_{*}f^{\prime*}(E,\vartheta)$.

2.8. Fourier–Mukai transforms of Hitchin pairs

Let $X$ and $Y$ be varieties, and let $Z\subset X\times Y$ be a closed sub-scheme. We denote by $\operatorname{\mathsf{pr}}_{X}\colon X\times Y\to X$ and $\operatorname{\mathsf{pr}}_{Y}\colon X\times Y\to Y$ the projections to the factors, and write $p\colon Z\to X$ and $q\colon Z\to Y$ for the restrictions of these projections to $Z$. In the following, we will always assume that $q\colon Z\to Y$ is flat and finite. For a coherent sheaf $\mathcal{P}\in\operatorname{\mathsf{Coh}}(Z)$, we define the Hitchin enhanced Fourier–Mukai transform $\Phi^{Z}_{\mathcal{P}}\colon\operatorname{\mathsf{Hi}}^{V}(X)\to\operatorname{\mathsf{Hi}}^{q_{*}p^{*}V}(Y)$ as the following composition of the three functors discussed in Subsections 2.5, 2.6, and 2.7:

[TABLE]

Remark \theremark.

Note that, in contrast to the usual convention for Fourier–Mukai transforms, none of our functors are derived. Hence, the functor $\Phi^{Z}_{\mathcal{P}}\colon\operatorname{\mathsf{Hi}}^{V}(X)\to\operatorname{\mathsf{Hi}}^{q_{*}p^{*}V}(Y)$ needs not to be exact. However, it is exact whenever $\mathcal{P}$ is flat over $X$, as will be the case in all our applications; compare [Kru18a, Thm. 1.1]. In this case, the Hitchin enhanced Fourier–Mukai transform is compatible with the usual Fourier–Mukai transform $\Phi_{\mathcal{P}}\colon\operatorname{\mathsf{Coh}}(X)\to\operatorname{\mathsf{Coh}}(Y)$ in the sense that we have

[TABLE]

where $\operatorname{\mathsf{For}}_{Y}\colon\operatorname{\mathsf{Hi}}^{q_{*}p^{*}V}(Y)\to\operatorname{\mathsf{Coh}}(Y)$ and $\operatorname{\mathsf{For}}_{X}\colon\operatorname{\mathsf{Hi}}^{V}(X)\to\operatorname{\mathsf{Coh}}(X)$ are the forgetful functors.

Example \theexample.

Let $f\colon Y\to X$ be a morphism, and let $\Gamma_{f}=(f\times\operatorname{\mathsf{id}}_{Y})(Y)\subset X\times Y$ be its graph. Then, for every $L\in\operatorname{\mathsf{Coh}}(Y)$ and $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$, we have a natural isomorphism $\Phi^{\Gamma_{f}}_{(f\times\operatorname{\mathsf{id}}_{Y})_{*}L}\cong f^{*}(E,\vartheta)\otimes L$. In particular, $\Phi^{\Gamma_{f}}_{\mathcal{O}_{\Gamma_{f}}}(E,\vartheta)\cong f^{*}(E,\vartheta)$.

Lemma \thelemma.

Let $0\to\mathcal{P}^{\prime}\to\mathcal{P}\to\mathcal{P}^{\prime\prime}\to 0$ be a short exact sequence of coherent sheaves on $Z$. Then, for every Hitchin bundle $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$, there is the following short exact sequence in $\operatorname{\mathsf{Hi}}^{q^{\prime}_{*}p^{\prime*}V}(Y)$:

[TABLE]

Proof.

This follows from the fact that tensor products by vector bundles are exact, together with the fact that a short exact sequence of Hitchin pairs is given by a short exact sequence of the underlying coherent sheaves. ∎

Let now $i\colon Z^{\prime}\hookrightarrow Z$ be a closed embedding, and let

[TABLE]

be the restrictions of the projections to $Z^{\prime}$. The canonical surjection $\mathcal{O}_{Z}\to\mathcal{O}_{Z^{\prime}}$ induces a surjection $\varphi\colon q_{*}p^{*}V\to q_{*}^{\prime}p^{\prime*}V$. Recall that this yields a functor $\varphi^{\#}\colon\operatorname{\mathsf{Hi}}^{q_{*}^{\prime}p^{\prime*}V}(X)\to\operatorname{\mathsf{Hi}}^{q^{*}p_{*}V}(X)$; see Subsection 2.4.

Lemma \thelemma.

For $\mathcal{Q}\in\operatorname{\mathsf{Coh}}(Z^{\prime})$ there is an isomorphism of functors $\Phi_{i_{*}\mathcal{Q}}^{Z}\cong\varphi^{\#}\circ\Phi_{\mathcal{Q}}^{Z^{\prime}}$.

Proof.

This follows by the projection formula for the embedding $i\colon Z^{\prime}\hookrightarrow Z$, together with Subsection 2.4. ∎

Let now $f\colon T\to Y$ be a morphism of schemes, and let $Z_{T}:=T\times_{Y}Z\subset T\times X$ be the fiber product such that we have a Cartesian diagram

[TABLE]

Lemma \thelemma.

There is an isomorphism of functors $f^{*}\circ\Phi^{Z}_{\mathcal{P}}\cong\Phi^{Z_{T}}_{f^{\prime*}\mathcal{P}}$.

Proof.

This follows from Subsection 2.7. ∎

2.9. Stability of Hitchin pairs

For this subsection, let $X$ be a smooth projective variety of dimension $n$. We fix an ample numerical class $H\in\operatorname{\mathsf{N}}^{1}(X)$. For every sheaf $A\in\operatorname{\mathsf{Coh}}(X)$, we define its $H$-degree and its $H$-slope by

[TABLE]

A Hitchin bundle $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$ is called ($H$-slope) stable if for every Hitchin subsheaf $A\subset E$ with $\operatorname{\mathsf{rank}}(A)<\operatorname{\mathsf{rank}}(E)$, we have $\mu(A)<\mu(E)$. It is called ($H$-slope) semi-stable if for every Hitchin subsheaf $A\subset E$, we have $\mu(A)\leq\mu(E)$.

Clearly, if $E$ is (semi-)stable as an ordinary sheaf, the Hitchin pair $(E,\vartheta)$ is (semi-)stable too. However, the converse is not true.

Example \theexample.

Let $X$ be a smooth projective curve, and let $L\in\operatorname{\mathsf{Pic}}X$ be a line bundle of degree $-1$. We set $E:=\mathcal{O}_{X}\oplus L$. Then $E$ is not stable and has exactly one destabilizing subsheaf, namely the direct summand $\mathcal{O}_{X}$. Now, let

[TABLE]

and consider the $L$-cotwisted Hitchin pair $(E,\vartheta)$. Then, the subbundle $\mathcal{O}_{X}\subset E$ is not a Hitchin subbundle, hence $(E,\vartheta)$ is stable.

Remark \theremark.

Let $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$ be stable (or semi-stable), and let $L\in\operatorname{\mathsf{Pic}}X$. Then $(E,\vartheta)\otimes L$ is again stable (or semi-stable).

Lemma \thelemma.

Let $\varphi\colon V\to Q$ be a surjective morphism of vector bundles on $X$. A Hitchin pair $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{Q}(X)$ is (semi-)stable if and only if $\varphi^{\#}(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$ is (semi-)stable.

Proof.

By Subsection 2.4, $A\subset E$ is a Hitchin subsheaf of $(E,\vartheta)$ if and only if it is a Hitchin subsheaf of $\varphi^{\#}(E,\vartheta)$. The assertion now follows from the fact that the slope of a Hitchin pair is defined as the slope of the underlying sheaf. ∎

Proposition \theproposition.

For every $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$, there is a unique filtration, called the Harder–Narasimhan filtration of $(E,\vartheta)$, by Hitchin subsheaves

[TABLE]

such that all the successive quotients $(\operatorname{\mathsf{F}}^{i}E/\operatorname{\mathsf{F}}^{i+1}E,\bar{\vartheta})$ are semi-stable Hitchin pairs with $\mu(\operatorname{\mathsf{F}}^{i}E/\operatorname{\mathsf{F}}^{i+1}E)<\mu(\operatorname{\mathsf{F}}^{i+1}E/\operatorname{\mathsf{F}}^{i+2}E)$ for every $i=0,\dots,m-2$.

Proof.

See [Sim94, Sect. 3]. ∎

Remark \theremark.

The Harder–Narasimhan filtration of the Hitchin pair $(E,\vartheta)$ does not need to agree with the Harder–Narasimhan filtration of $E$; compare Subsection 2.9.

2.10. Equivariant Hitchin pairs

Let $G$ be a finite group acting on a scheme $X$. A $G$-equivariant vector bundle $(V,\alpha)$ on $X$ consists of a vector bundle $V\in\operatorname{\mathsf{VB}}(X)$ and a family of isomorphisms $\alpha=\{\alpha_{g}\colon V\xrightarrow{\sim}g^{*}V\}$, called a $G$-linearization, such that for all $g,h\in G$ the following diagram commutes:

[TABLE]

Remark \theremark.

Let $(V,\alpha)$ be a $G$-equivariant vector bundle. For $g\in G$, we consider the composition

[TABLE]

which is an autoequivalence of $\operatorname{\mathsf{Hi}}^{V}(X)$. In the following, we simply denote this autoequivalence by $g^{*}\colon\operatorname{\mathsf{Hi}}^{V}(X)\xrightarrow{\sim}\operatorname{\mathsf{Hi}}^{V}(X)$. For $g,h\in G$, there is a canonical ismorphism of autoequivalences $g^{*}h^{*}\cong(hg)^{*}$, which means that we have an action of the group $G$ on the category $\operatorname{\mathsf{Hi}}^{V}(X)$.

Given a $G$-equivariant vector bundle $(V,\alpha)$, a $G$-equivariant $V$-cotwisted Hitchin pair \bigl{(}(E,\vartheta),\lambda\bigr{)} consists of a Hitchin pair $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$ and a family of isomorphisms of Hitchin pairs $\lambda=\{\lambda_{g}\colon(E,\vartheta)\xrightarrow{\sim}g^{*}(E,\vartheta)\}$ such that for all $g,h\in G$ the following diagram commutes:

[TABLE]

That $\lambda_{g}$ is a morphism of Hitchin pairs means explicitly that the following diagram commutes:

[TABLE]

A $G$-equivariant Hitchin subsheaf of \bigl{(}(E,\vartheta),\lambda\bigr{)} is a Hitchin subsheaf $A\subset E$ such that for every $g\in G$, we have the equality $\lambda_{g}(A)=g^{*}A$ of subsheaves of $g^{*}E$.

Let $\pi\colon X\to X/G$ be a geometric quotient. For every $g\in G$, we have the equality $\pi\circ g=\pi$, which gives an isomorphism of functors $g^{*}\circ\pi^{*}\cong\pi^{*}$. Hence, for every $V\in\operatorname{\mathsf{VB}}(X/G)$, the pull-back $\pi^{*}V$ carries a canonical $G$-linearisation. Furthermore, for every Hitchin pair $(F,\eta)\in\operatorname{\mathsf{Hi}}^{V}(X/G)$, the pull-back $\pi^{*}(F,\eta)$ has canonically the structure of a $G$-equivariant $\pi^{*}V$-cotwisted Hitchin pair.

Lemma \thelemma.

Let a finite group $G$ act on a smooth projective variety $X$ such that $Y=X/G$ is again smooth and $\pi\colon X\to Y$ is flat. Let $H\in\operatorname{\mathsf{N}}^{1}(Y)$ be an ample class, $V\in\operatorname{\mathsf{VB}}(Y)$, and $(F,\eta)\in\operatorname{\mathsf{Hi}}^{V}(Y)$ a Hitchin bundle.

1. (i)

If $\mu_{\pi^{*}H}(A)\leq\mu_{\pi^{*}H}(\pi_{n}^{*}F)$ holds for all $\mathfrak{S}_{n}$-equivariant Hitchin subsheaves $A$ of $\pi^{*}(F,\eta)$ with $\operatorname{\mathsf{rank}}A<\operatorname{\mathsf{rank}}F$, then $F$ is slope semi-stable with respect to $H$. 2. (ii)

If $\mu_{\pi^{*}H}(A)<\mu_{\pi^{*}H}(\pi_{n}^{*}F)$ holds for all $\mathfrak{S}_{n}$-equivariant Hitchin subsheaves $A$ of $\pi^{*}(F,\eta)$ with $\operatorname{\mathsf{rank}}A<\operatorname{\mathsf{rank}}F$, then $F$ is slope stable with respect to $H$.

Proof.

The proof is completely analogous to the proof for vector bundles without the structure of a Hitchin pair; see [Kru18b, Lem. 1.1]. ∎

3. Construction of Hitchin pairs on Hilbert schemes of points

3.1. Hilbert schemes of points

Let $X$ be a smooth quasi-projective variety. For every $n\in\mathbb{N}$ there is a fine moduli space $X^{[n]}$ of zero-dimensional sub-schemes of $X$ of length $n$, called the Hilbert scheme of $n$ points on $X$ (also called punctual Hilbert scheme). This means that there is a closed sub-scheme $\Xi_{n}\subset X\times X^{[n]}$ which is flat and finite of degree $n$ over $X^{[n]}$, called the universal family, such that, for every scheme $T$ and every closed sub-scheme $Z\subset X\times T$ which is flat and finite of degree $n$ over $T$, there is a classifying morphism $f\colon T\to X^{[n]}$ such that $Z=(\operatorname{\mathsf{id}}_{X}\times f)^{-1}\Xi_{n}$, where $(\operatorname{\mathsf{id}}_{X}\times f)^{-1}\Xi_{n}\cong T\times_{X^{[n]}}\Xi_{n}$ is the scheme-theoretic preimage.

There is the Hilbert–Chow morphism $\mu\colon X^{[n]}\to X^{(n)}$ to the symmetric product $X^{(n)}:=X^{n}/\mathfrak{S}_{n}$ that sends any zero-dimensional sub-scheme $\xi\subset X$ to its weighted support: $\mu([\xi])=\sum_{x\in\xi}\ell(\xi,x)\cdot x$.

If $X=C$ is a smooth curve, $\mu$ is an isomorphism, hence $C^{[n]}\cong C^{(n)}$. If $X$ is a smooth surface, $X^{[n]}$ is smooth and $\mu\colon X^{[n]}\to X^{(n)}$ is a crepant resolution of the singularities of the symmetric product.

3.2. Tautological bundles and Hitchin pairs

Let $X$ be a smooth quasi-projective variety, let $n\in\mathbb{N}$, and let $X\xleftarrow{p}\Xi_{n}\xrightarrow{q}X^{[n]}$ be the projections from the universal family of the Hilbert scheme $X^{[n]}$. Given a sheaf $E\in\operatorname{\mathsf{Coh}}(X)$, the associated tautological sheaf is defined by $E^{[n]}=q_{*}p^{*}E\in\operatorname{\mathsf{Coh}}(X^{[n]})$. Over a point $[\xi]\in X^{[n]}$ corresponding to a zero-dimensional sub-scheme $\xi\subset X$ of length $n$, the fiber of the tautological sheaf is given by $E^{[n]}([\xi])=\operatorname{\mathsf{H}}^{0}(E_{\mid\xi})$. If $E$ is a vector bundle, then $E^{[n]}$ is again a vector bundle with $\operatorname{\mathsf{rank}}E^{[n]}=n\operatorname{\mathsf{rank}}E$. This follows from the fact that $q\colon\Xi_{n}\to X^{[n]}$ is flat and finite of degree $n$.

Let $V\in\operatorname{\mathsf{VB}}(X)$. If $E$ carries the structure of a $V$-cotwisted Hitchin pair, we get an induced structure of a $V^{[n]}$-cotwisted Hitchin pair on $E^{[n]}$. More precisely, for $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$, we define the associated tautological Hitchin pair, using the Hitchin enhanced Fourier–Mukai transform as introduced in Subsection 2.8, by

[TABLE]

Remark \theremark.

To provide some intuition, let us give a concrete description of the induced map $\vartheta^{\{n\}}=q_{*}p^{*}\vartheta\circ\nu_{q,E,V}\colon E^{[n]}\otimes V^{[n]}\to E^{[n]}$. For a zero-dimensional sub-scheme $\xi\subset X$ of length $n$, the fiber of $\vartheta^{\{n\}}$ over the point $[\xi]\in X^{[n]}$ is given by the composition

[TABLE]

The fact that \bigl{(}\vartheta^{\{n\}}\circ(\vartheta^{\{n\}}\otimes\operatorname{\mathsf{id}}_{V^{[n]}})\bigr{)}_{\mid E^{[n]}\otimes\bigwedge^{2}V^{[n]}}=0, which already follows from Subsection 2.7, can also be read off from the above description of $\vartheta^{\{n\}}$.

Lemma \thelemma.

Let $T$ be a scheme, let $Z\subset X\times T$ be a sub-scheme which is flat and finite of degree $n$ over $T$, and let $f\colon T\to X^{[n]}$ be the classifying morphism for $Z$. Then, for $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$, we have a natural isomorphism of $f^{*}V^{[n]}$-cotwisted Hitchin pairs

[TABLE]

Proof.

This follows by applying Subsection 2.8 to the Cartesian diagram

[TABLE]

∎

3.3. Tangent bundle of the Hilbert scheme

If $X$ is a smooth curve or surface, the Hilbert scheme $X^{[n]}$ is again smooth. In this case, one might hope that, if we start with a sheaf $E\in\operatorname{\mathsf{Coh}}(X)$ and a non-zero Higgs field $E\to E\otimes\Omega_{X}$, we get an induced non-zero Higgs field $E^{[n]}\to E^{[n]}\otimes\Omega_{X^{[n]}}$ on $E^{[n]}$. However, as the following example shows, there cannot be such a general construction.

Example \theexample.

Let $X=\mathbb{P}^{1}$. Then, as $\Omega_{X}=\mathcal{O}_{\mathbb{P}^{1}}(-2)$, the bundle $E:=\mathcal{O}_{\mathbb{P}^{1}}(-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(1)$ has a non-zero Higgs field. There is an isomorphism $(\mathbb{P}^{1})^{[2]}\cong(\mathbb{P}^{1})^{(2)}\cong\mathbb{P}^{2}$ under which the tautological bundles are given by

[TABLE]

see [Nag17, Sect. 3]. Hence, $E^{[2]}\cong\mathcal{O}_{\mathbb{P}^{2}}(-1)^{\oplus 2}\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus 2}$. Using the Euler sequence, one computes $\operatorname{\mathsf{H}}^{0}(\Omega_{\mathbb{P}^{2}}(-1))=\operatorname{\mathsf{H}}^{0}(\Omega_{\mathbb{P}^{2}})=\operatorname{\mathsf{H}}^{0}(\Omega_{\mathbb{P}^{2}}(1))=0$. It follows that $\operatorname{\mathsf{Hom}}(E^{[2]},E^{[2]}\otimes\Omega_{\mathbb{P}^{2}})=0$, which means that there is no non-zero Higgs field on $E^{[2]}$, though there is one on $E$.

However, our construction from the previous subsection equips $E^{[n]}$ with the structure of a logarithmic Higgs sheaf, with respect to the boundary divisor $B_{n}\subset X^{[n]}$, whenever there is a Higgs field on $E$. Indeed, let $(E,\vartheta)$ be a Higgs sheaf, i.e., a $\operatorname{\mathsf{T}\!}_{X}$-cotwisted Hitchin pair. Then $(E,\vartheta)^{[n]}$ is a $(\operatorname{\mathsf{T}\!}_{X})^{[n]}$-cotwisted Hitchin pair. Hence, our assertion follows from the following result.

Theorem 3.1** ([Sta16]).**

Let $X$ be a smooth curve or surface. Then there is an isomorphism

[TABLE]

Proof.

For $X$ a smooth surface, this is [Sta16, Thm. B]. The proof in the case case where $X$ is a smooth curve is essentially the same. The only difference is that we do not need to restrict to an open subset $U\subset X^{[n]}$ (as done in the surface case on top of page 1181 of [Sta16].) since, in the curve case, the universal family $\Xi_{n}$ is already smooth everywhere (for $X$ a smooth curve, we have $\Xi_{n}\cong X\times X^{(n-1)}$). ∎

4. Reconstruction in the case of curves

4.1. Curves of genus $g\geq 2$

In this subsection, we will prove Theorem 1.1. So let $C$ be a smooth projective curve of genus $\operatorname{\mathsf{g}}(C)\geq 2$, let $V\in\operatorname{\mathsf{VB}}(C$), and let $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(C)$ be a semi-stable Hitchin bundle. We enhance the reconstruction method of [BN12] and [BN17, Sect. 2], which reconstructs vector bundles $E$ on $C$ from their tautological images $E^{[n]}$ on the symmetric product $C^{(n)}$, to Hitchin bundles.

Let $C\cong\Delta\subset C\times C$ be the diagonal with ideal sheaf ${\mathcal{I}}_{\Delta}$. We denote the $(n-1)$-th infinitesimal thickening of the diagonal by $Z:=V({\mathcal{I}}_{\Delta}^{n})\subset C\times C$. Via the projection $\operatorname{\mathsf{pr}}_{1}\colon C\times C\to C$, the sub-scheme $Z\subset C\times C$ is a flat family of length $n$ sub-schemes of $C$ over $C$. Hence, we get a classifying morphism $\iota\colon C\to C^{(n)}$, which is a closed embedding with image the small diagonal in $C^{(n)}$. By Subsection 3.2, we have

[TABLE]

Hence, it is sufficient to show that the isomorphism class of $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(C)$ is determined by $\Phi^{Z}_{\mathcal{O}_{Z}}(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{b_{*}a^{*}V}(C)$, where $a,b\colon Z\to C$ are the restrictions of the projections $\operatorname{\mathsf{pr}}_{1},\operatorname{\mathsf{pr}}_{2}\colon C\times C\to C$.

The structure sheaf $\mathcal{O}_{Z}$ has the filtration

[TABLE]

where ${\mathcal{J}}_{\Delta}:={\mathcal{I}}_{\Delta}/{\mathcal{I}}_{\Delta}^{n}$. This induces a filtration by Hitchin subsheaves

[TABLE]

which we will show to be the Harder–Narasimhan filtration of $\Phi^{Z}_{\mathcal{O}_{Z}}(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{b_{*}a^{*}V}(C)$. Note that we have short exact sequences

[TABLE]

where $i\colon C\hookrightarrow Z$ is the closed embedding of the reduced diagonal. By Subsection 2.8 combined with Subsection 2.8, we have \Phi^{Z}_{i_{*}\omega_{C}^{\otimes k}}\cong\varphi^{\#}\bigl{(}(E,\vartheta)\otimes\omega_{C}^{\otimes k}\bigr{)} where $\varphi\colon b_{*}a^{*}V\twoheadrightarrow V$ is the canonical surjection induced by $\mathcal{O}_{Z}\twoheadrightarrow\mathcal{O}_{\Delta}$. Hence, applying Subsection 2.8 to (11) gives a short exact sequence

[TABLE]

By the assumption that $(E,\vartheta)$ is semi-stable, Subsection 2.9, and Subsection 2.9, we see that the factor \Phi^{Z}_{{\mathcal{J}}_{\Delta}^{k}}(E,\vartheta)/\Phi^{Z}_{{\mathcal{J}}_{\Delta}^{k+1}}(E,\vartheta)\cong\varphi^{\#}\bigl{(}(E,\vartheta)\otimes\omega_{C}^{\otimes k}\bigr{)} is semi-stable. Furthermore, since $\operatorname{\mathsf{deg}}(\omega_{C})>0$ by the assumption on the genus $\operatorname{\mathsf{g}}(C)$, we have strict inequalities

[TABLE]

This means that (10) is indeed the Harder–Narasimhan filtration of $\Phi^{Z}_{\mathcal{O}_{Z}}(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{b_{*}a^{*}V}(X)$; see Subsection 2.9. The uniqueness of the first factor $\varphi^{\#}(E,\vartheta)$ of the Harder–Narasimhan filtration, together with Subsection 2.4(i), now show that the isomorphism class of $(E,\vartheta)$ is determined by $\Phi^{Z}_{\mathcal{O}_{Z}}(E,\vartheta)\cong\iota^{*}(E,\vartheta)^{[n]}$.

Remark \theremark.

In analogy with [BN17, Sect. 2], we can relax the assumptions of Theorem 1.1 as follows. Let $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$ be a Hitchin bundle with Harder–Narasimhan filtration

[TABLE]

We set $\mu_{\mathsf{min}}(E,\vartheta):=\mu(E/\operatorname{\mathsf{F}}^{1}E)$ and $\mu_{\mathsf{max}}(E,\vartheta):=\mu(\operatorname{\mathsf{F}}^{m-1}E)$. Then, instead of assuming in Theorem 1.1 that $(E,\vartheta)$ and $(F,\eta)$ are semi-stable, it is sufficient to assume that

[TABLE]

4.2. Elliptic curves

In this subsection, we prove Theorem 1.2 as a consequence of the following more general result; compare [Kru18a, Thm. 5.2].

Proposition \theproposition.

Let $X$ be a smooth projective variety which has $n+1$ automorphisms $\sigma_{0},\dots,\sigma_{n}\in\operatorname{\mathsf{Aut}}(X)$ with empty pairwise equalizers, which means that $\sigma_{i}(x)\neq\sigma_{j}(x)$ for all $i\neq j$ and all $x\in X$. Then, for every $n\in\mathbb{N}$ and every two Hitchin pairs $(E,\vartheta),(F,\eta)\in\operatorname{\mathsf{Hi}}^{V}(X)$, we have

[TABLE]

Proof.

Replacing $\sigma_{i}$ by $\sigma_{0}^{-1}\circ\sigma_{i}$, we can assume without loss of generality that $\sigma_{0}=\operatorname{\mathsf{id}}_{X}$. By the assumption on the automorphisms, the graphs $\Gamma_{\sigma_{i}}$ are pairwise disjoint. Hence, for every $j\in\{0,\dots,n\}$, the union

[TABLE]

is flat and finite of degree $n$ over $X$. Let $f_{j}\colon X\to X^{[n]}$ be the classifying morphism for $G_{j}$. By Subsection 3.2, we have $f_{j}^{*}(E,\vartheta)^{[n]}\cong\Phi^{G_{j}}_{\mathcal{O}_{G_{j}}}(E,\vartheta)$. Since the $\Gamma_{\sigma_{j}}$ are disjoint, we have $\mathcal{O}_{G_{j}}\cong\oplus_{i\neq j}\mathcal{O}_{\Gamma_{\sigma_{i}}}$. Together with Subsection 2.8 and Subsection 2.8, this gives

[TABLE]

where $\varphi^{j}_{i}\colon\Phi_{\mathcal{O}_{G_{j}}}(V)\cong\oplus_{i\neq j}\sigma_{i}^{*}V\to\Phi_{\mathcal{O}_{\Gamma_{\sigma_{i}}}}(V)\cong\sigma_{i}^{*}V$ is the projection, which is induced by the surjection $\mathcal{O}_{G_{j}}\to\mathcal{O}_{\Gamma_{\sigma_{i}}}$. We also set $\mathbb{V}:=\oplus_{i=1}^{n}\sigma_{i}^{*}V$, and write

[TABLE]

for the appropriate projections to the summands. Applying $\psi_{j}^{\#}\colon\operatorname{\mathsf{Hi}}^{\oplus_{i\neq j}\sigma_{i}^{*}V}(X)\to\operatorname{\mathsf{Hi}}^{\mathbb{V}}(X)$ to (12) gives an isomorphism

[TABLE]

Here, the $\sigma_{i}^{*}$ on the right-hand side is a shortcut for the autoequivalence $\alpha_{\sigma_{i}}^{\#}\circ\sigma_{i}^{*}$ of $\operatorname{\mathsf{Hi}}^{\mathbb{V}}(X)$ where $\alpha_{\sigma_{i}}\colon\mathbb{V}\to\sigma_{i}^{*}\mathbb{V}$ is the canonical isomorphism given by permutation of the summands; compare Subsection 2.10.

Considering these isomorphisms for varying $j=0,\dots,n$ gives

[TABLE]

Since the category $\operatorname{\mathsf{Hi}}^{\mathbb{V}}(X)$ is Krull-Schmidt, [Kru18a, Prop. 5.4] gives

[TABLE]

Combining implication (13) with implication (14) and Subsection 2.4(i) proves the assertion. ∎

Theorem 1.2 is a special case of Subsection 4.2 since, on an elliptic surface, there is an infinite group of automorphisms with empty pairwise equalizers, namely the transpositions.

5. Stability in the case of curves

In this section, we prove Theorem 1.4. The special case of bundles without the structure of a Hitchin pair was proved in [Kru18b], and here we explain the extra ingredients necessary for the same proof to work for Hitchin pairs.

Let $C$ be a smooth projective curve, and let $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(C)$ be a stable (or semi-stable) Hitchin pair with $\mu(E)\notin[-1,n-1]$ (or $\mu(E)\notin(-1,n-1)$). Let $\pi_{n}\colon C^{n}\to C^{(n)}$ be the $\mathfrak{S}_{n}$-quotient morphism. As discussed in Subsection 2.10, the pull-back $\pi_{n}^{*}(E,\vartheta)^{[n]}\in\operatorname{\mathsf{Hi}}^{\pi_{n}^{*}V^{[n]}}(C^{n})$ of the tautological Hitchin bundle carries a canonical $\mathfrak{S}_{n}$-linearization. By Subsection 2.10, for $(E,\,\vartheta)^{[n]}$ to be stable (or semi-stable), it is sufficient to prove that, for every $\mathfrak{S}_{n}$-equivariant Hitchin subsheaf $A\subset\pi_{n}^{*}E^{[n]}$ with $\operatorname{\mathsf{rank}}A<\operatorname{\mathsf{rank}}(\pi_{n}^{*}E^{[n]})\,=\,n\operatorname{\mathsf{rank}}(E)$, we have the inequality $\mu_{\widetilde{H}_{n}}(A)\,<\,\mu_{\widetilde{H}_{n}}(\pi^{*}E^{[n]})$ (or $\mu(A)\,\leq\,\mu(\pi^{*}E^{[n]})$).

The key to the proof of the stability criterion for tautological bundles in [Kru18b] is the existence of a short exact sequence

[TABLE]

where $\operatorname{\mathsf{pr}}_{1}\colon C^{n}\to C$ and $\overline{\operatorname{\mathsf{pr}}}_{1}\colon C^{n}\to C^{n-1}$ are the projections to the first factor and to the last $n-1$ factors, respectively, and $\delta_{n}(1)=\sum_{i=2}^{n}\Delta_{1i}$ where

[TABLE]

see [Kru18b, Prop 1.5]. Let now $A\subset\pi^{*}_{n}E^{[n]}$ be an equivariant Hitchin subsheaf. Setting $A^{\prime}:=i_{E}^{-1}A\subset\operatorname{\mathsf{pr}}_{1}^{*}E(-\delta_{n}(1))$ and $A^{\prime\prime}:=p_{E}(A)$, we get a commutative diagram

[TABLE]

with exact columns and rows. The proof in [Kru18b] uses the (semi-)stability of $E$ (by assumption) and the (semi-)stability of $E^{[n-1]}$ (by induction) to find upper bounds for the slopes of $A^{\prime}$ and $A^{\prime\prime}$, which lead to the desired inequality $\mu_{\widetilde{H}_{n}}(A)<\mu_{\widetilde{H}_{n}}(\pi^{*}E^{[n]})$ (or $\mu(A)\leq\mu(\pi^{*}E^{[n]})$). However, we only know the semi-stability of $(E,\vartheta)$ as a Hitchin pair, which does not imply the stability of $E$ as an ordinary vector bundle; see Subsection 2.9. The reason that, nevertheless, the proof of loc. cit. works for Hitchin pairs is that we can enhance (15) to a short exact sequence of $\pi_{n}^{*}V^{[n]}$-cotwisted Hitchin pairs.

For this, we need to have a look at the construction of the short exact sequence (15) in the proof of [Kru18b, Prop 1.5]. We have $(\pi_{n}\times\operatorname{\mathsf{id}}_{C})^{-1}\Xi_{n}=D_{n}$, where $D_{n}=\bigcup_{k=1}^{n}\Gamma_{k}\subset C^{n}\times C$ is the union of the graphs $\Gamma_{k}:=\Gamma_{\operatorname{\mathsf{pr}}_{k}}$ of the projections $\operatorname{\mathsf{pr}}_{k}\colon C^{n}\to C$ to the $k$-th factor. It follows by flat base change that $E^{[n]}\cong\Phi_{\mathcal{O}_{D_{n}}}(E)$. Under this isomorphism, (15) is induced by the canonical short exact sequence of Fourier–Mukai kernels

[TABLE]

together with the isomorphism $\mathcal{O}_{\cup_{k=2}^{n}\operatorname{\mathsf{pr}}_{k}}\cong(\overline{\operatorname{\mathsf{pr}}}_{1}\times\operatorname{\mathsf{id}}_{C})^{*}\mathcal{O}_{D_{n-1}}$. In other words, the short exact sequence (15) is isomorphic to the exact sequence

[TABLE]

By Subsection 2.8 we get an enhanced version of (24) in the form of a short exact sequence

[TABLE]

of $\pi_{n}^{*}V^{[n]}$-cotwisted Hitchin pairs. The surjections $\mathcal{O}_{D_{n}}\to\mathcal{O}_{\Gamma_{1}}$ and $\mathcal{O}_{D_{n}}\to\mathcal{O}_{\cup_{k=2}^{n}\operatorname{\mathsf{pr}}_{k}}\cong(\overline{\operatorname{\mathsf{pr}}}_{i}\times\operatorname{\mathsf{id}}_{C})^{*}\mathcal{O}_{D_{n-1}}$ induce surjections

[TABLE]

Applying Subsection 2.8 together with Subsection 2.8 to the first term of (25), Subsection 2.8 (recalling the isomorphism $(\pi_{n}\times\operatorname{\mathsf{id}}_{C})^{*}\mathcal{O}_{\Xi_{n}}\cong\mathcal{O}_{D_{n}}$) to the second term of (25), and Subsection 2.8 (recalling the isomorphism $\mathcal{O}_{\cup_{k=2}^{n}\operatorname{\mathsf{pr}}_{k}}\cong(\overline{\operatorname{\mathsf{pr}}}_{i}\times\operatorname{\mathsf{id}}_{C})^{*}\mathcal{O}_{D_{n-1}}$) together with Subsection 2.8 to the third term of (25), we see that (25) is isomorphic to

[TABLE]

Lemma \thelemma.

Let, as above, $A$ be an $\mathfrak{S}_{n}$-equivariant Hitchin subsheaf of $\pi^{*}_{n}(E,\vartheta)^{[n]}$, and let $A^{\prime}:=i_{E}^{-1}(A)\subset\operatorname{\mathsf{pr}}_{1}^{*}E(-\delta_{n}(1))$ and $A^{\prime\prime}:=p_{E}(A)\subset\overline{\operatorname{\mathsf{pr}}}_{1}^{*}\pi_{n-1}^{*}E^{[n-1]}$.

1. (i)

$A^{\prime}$* is a Hitchin subsheaf of the $\operatorname{\mathsf{pr}}_{1}^{*}V$-cotwisted Hitchin pair $\operatorname{\mathsf{pr}}_{1}^{*}(E,\vartheta)\otimes\mathcal{O}(-\delta_{n}(1))$.* 2. (ii)

$A^{\prime\prime}$* is a Hitchin subsheaf of the $\overline{\operatorname{\mathsf{pr}}}_{1}^{*}\pi_{n-1}^{*}V^{[n-1]}$-cotwisted Hitchin pair $\overline{\operatorname{\mathsf{pr}}}_{1}^{*}\pi_{n-1}^{*}(E,\vartheta)^{[n-1]}$.*

Proof.

Since we now have the short exact sequence of Hitchin pairs (26), this follows directly from Subsection 2.2 and Subsection 2.4(iii). ∎

Now, as in [Kru18b, Subsect. 2.2], we choose the points $x_{2},\dots,x_{n}$ in such a way that all rows and columns of the diagram (22) stay exact after pull-back along the embedding

[TABLE]

Since $\iota^{*}\mathcal{O}_{C^{n}}(-\delta_{n}(1))\cong\mathcal{O}_{C}(-\sum_{i=2}^{n}x_{i})$ and $\operatorname{\mathsf{pr}}_{1}\circ\iota=\operatorname{\mathsf{id}}_{C}$, we have an isomorphism of $\iota^{*}\operatorname{\mathsf{pr}}_{1}^{*}V\cong V$-cotwisted Hitchin pairs

[TABLE]

By Section 5(i), $\iota^{*}A^{\prime}$ is a Hitchin subsheaf of $(E,\vartheta)\otimes\mathcal{O}(-\sum_{i=2}^{n}x_{i})$. Since, by assumption, $(E,\vartheta)$ is stable (or semi-stable), so is $(E,\vartheta)\otimes\mathcal{O}(-\sum_{i=2}^{n}x_{i})$; see Subsection 2.9. Hence, we have $\mu(\iota^{*}A^{\prime})<\mu(E(-\sum_{i=2}^{n}x_{i}))$ (or $\mu(\iota^{*}A^{\prime})\leq\mu(E(-\sum_{i=2}^{n}x_{i}))$). Now, in the case $\mu(E)>n-1$ (or $\mu(E)\geq n-1$), we can follow the computations of [Kru18b, Subsect. 2.2] to see that $\mu_{\widetilde{H}_{n}}(A)<\mu_{\widetilde{H}_{n}}(\pi_{n}^{*}E^{[n]})$ (or $\mu_{\widetilde{H}_{n}}(A)\leq\mu_{\widetilde{H}_{n}}(\pi_{n}^{*}E^{[n]})$) which, as discussed above, by Subsection 2.10 implies that $E^{[n]}$ is stable (or semi-stable).

If $\mu(E)\leq-1$, as in [Kru18b, Subsect. 2.3], we consider the closed embedding

[TABLE]

where, again, $x\in C$ is chosen in such a way that all rows and columns of the diagram (22) stay exact after pull-back along $\iota^{*}$. Since $\overline{\operatorname{\mathsf{pr}}}_{1}\circ\iota=\operatorname{\mathsf{id}}_{C^{n-1}}$, we have an isomorphism of $\iota^{*}\overline{\operatorname{\mathsf{pr}}}_{1}^{*}\pi_{n-1}^{*}V^{[n-1]}\cong\pi_{n-1}^{*}V^{[n-1]}$-cotwisted Hitchin pairs

[TABLE]

By induction, we may assume that $\pi_{n-1}^{*}(E,\vartheta)^{[n-1]}$ is stable (or semi-stable). By Section 5(ii), $\iota^{*}A^{\prime\prime}$ is a Hitchin subsheaf of $\pi_{n-1}^{*}(E,\vartheta)^{[n-1]}$. Hence, we get that $\mu(\iota^{*}A^{\prime\prime})<\mu(\pi_{n-1}^{*}E^{[n-1]})$ (or $\mu(\iota^{*}A^{\prime\prime})\leq\mu(\pi_{n-1}^{*}E^{[n-1]})$). Now, the assertion in the case $\mu(E)<-1$ (or $\mu(E)\leq-1$) follows from the computations in [Kru18b, Subsect. 2.3].

6. Higher dimensions

6.1. Reconstruction for higher dimensions

In this section, we proof Theorem 1.3. A version of this statement for coherent sheaves without the structure of a Hitchin bundle was proved in [Kru18a, Thm. 3.2 & Rem. 3.5]. Since the arguments which allow to lift the results to Hitchin pairs are very similar to those of the previous two sections, we will restrict ourselves to a quiet terse explanation this time.

Recall that there is the Hilbert–Chow morphism $\mu\colon X^{[n]}\to X^{(n)}$ and the quotient morphism $\pi\colon X^{n}\to X^{(n)}$. We consider the open subsets

[TABLE]

$X^{(n)}_{0}:=\pi(X^{n}_{0})\subset X^{(n)}$, and $X^{[n]}_{0}:=\mu^{-1}(X^{(n)}_{0})\subset X^{[n]}$. This gives the following diagram with Cartesian squares

[TABLE]

where all the vertical arrows are open immersions, and $\mu_{0}$ is an isomorphism.

For $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$, we define the $\mathfrak{S}_{n}$-equivariant $(\oplus_{i=1}^{n}\operatorname{\mathsf{pr}}_{i}^{*}V)$-cotwisted Hitchin pair $\operatorname{\mathsf{C}}(E,\vartheta)=(\oplus_{i=1}^{n}\operatorname{\mathsf{pr}}_{i}^{*}E,\tilde{\vartheta},\lambda)$ by

[TABLE]

where the middle map is the projection to the appropriate factors, and $\lambda_{g}\colon\oplus_{i}\operatorname{\mathsf{pr}}_{i}^{*}E\to g^{*}(\oplus_{i}\operatorname{\mathsf{pr}}_{i}^{*}E)$ is the direct sum of the canonical isomorphisms $\operatorname{\mathsf{pr}}_{i}^{*}E\xrightarrow{\sim}g^{*}\operatorname{\mathsf{pr}}_{g(i)}^{*}E$.

Note that the open subset $X^{n}_{0}\subset X^{n}$ is stable under the $\mathfrak{S}_{n}$-action on $X^{n}$, hence the morphism $\pi_{0}\colon X^{n}_{0}\to X^{(n)}_{0}$ is a $\mathfrak{S}_{n}$-quotient. Thus, as explained in Subsection 2.10, the Higgs bundle $j_{*}\pi_{0}^{*}\mu_{0*}i^{*}(E,\vartheta)^{[n]}$ carries a canonical $\mathfrak{S}_{n}$-linearization.

Lemma \thelemma.

For every $(E,\vartheta)\in\operatorname{\mathsf{Hi}}^{V}(X)$ such that $E$ is reflexive, we have an isomorphism of $\mathfrak{S}_{n}$-equivariant $(\oplus_{i=1}^{n}\operatorname{\mathsf{pr}}_{i}^{*}V)$-cotwisted Hitchin pairs

[TABLE]

Proof.

For vector bundles without the structure of a Hitchin pair, this is [Sta16, Lem. 1.1]. For the straightforward generalization to reflexive sheaves, see [Kru18a, Sect. 3.1]. That the two structures of Hitchin pairs $\left(\bigoplus_{i=1}^{n}\operatorname{\mathsf{pr}}_{i}^{*}E\right)\otimes\left(\bigoplus_{i=1}^{n}\operatorname{\mathsf{pr}}_{i}^{*}V\right)\to\bigoplus_{i=1}^{n}\operatorname{\mathsf{pr}}_{i}^{*}E$ on both sides of the alleged isomorphism agree can be checked on the open dense subset $X^{n}_{0}\subset X^{n}$. This can be done either using the results on Hitchin enhanced Fourier–Mukai transforms of Subsection 2.8, or the concrete description of $\vartheta^{\{n\}}$ of Subsection 3.2. ∎

Proof of Theorem 1.3.

Let $\delta\colon X\hookrightarrow X^{n}$, $\delta(x)=(x,\dots,x)$ be the embedding of the small diagonal. We have $\delta^{*}\operatorname{\mathsf{C}}(E,\vartheta)\cong(E,\vartheta)^{\oplus n}$ with the induced $\mathfrak{S}_{n}$-linearization given by permutation of the direct summands. Hence, the $\mathfrak{S}_{n}$-invariants are given by \bigl{(}\delta^{*}\operatorname{\mathsf{C}}(E,\vartheta)\bigr{)}^{\mathfrak{S}_{n}}\cong(E,\vartheta). By Subsection 6.1, this implies

[TABLE]

which means that we can reconstruct the isomorphism class of $(E,\vartheta)$ form $(E,\vartheta)^{[n]}$. ∎

6.2. Stability of tautological Hitchin bundles on Hilbert schemes of points on

surfaces

Let $X$ be a smooth projective surface, and let $H$ be an ample divisor on $X$. There is a unique divisor $H_{(n)}$ on the symmetric product $X^{(n)}$ with $\pi_{n}^{*}H_{(n)}=\sum_{i=1}^{n}\operatorname{\mathsf{pr}}_{i}^{*}H$. We set $H_{[n]}:=\mu^{*}H_{(n)}$. This divisor is big and nef, but not ample, as it is trivial along the exceptional divisor of $\mu$. The definitions of stable vector bundles and Hitchin bundles still make sense for non-ample divisors. In [Sta16, Sect. 1], it is shown that, if $E$ is a $H$-slope stable vector bundle on $X$ with $E\not\cong\mathcal{O}_{X}$, the associated tautological bundle $E^{[n]}$ is $H_{[n]}$-slope stable. Using Subsection 6.1 while going through the proof of [Sta16], it is quite easy to see that this result also generalizes to Hitchin pairs: If $(E,\vartheta)$ is a $H$-slope stable Hitchin pair on $X$ with $E\not\cong\mathcal{O}_{X}$, then $(E,\vartheta)^{[n]}$ is $H_{[n]}$-slope stable.

In [Sta16], it is shown that in a neighborhood of $H_{[n]}$ in $\operatorname{\mathsf{N}}^{1}(X)$, there is also an ample class $I$ such that $E^{[n]}$ is $I$-slope stable if $E\not\cong\mathcal{O}_{X}$ is $H$-slope stable. It seems likely that also this result generalizes to Hitchin pairs, but we have not checked the details.

Acknowledgements

The first-named author wishes to thank Philipps-Universität Marburg for hospitality while this work was carried out. He also acknowledges a partial support of a J. C. Bose Fellowship.