The Spectral base and quotients of bounded symmetric domains by cocompact lattices

Let $X$ be a projective manifold, and denote by $\Omega_{X}^{1}$ the holomorphic cotangent bundle of $X$. We collect some basic definitions and facts about Higgs sheaves/bundles, which we refer [Sim92, BS94, BS09, LZZ17].

Given a Higgs sheaf $(\mathscr{E},\varphi)$ over a projective manifold $X$, a coherent subsheaf $\mathscr{F}\subset\mathscr{E}$ is said to be $\varphi$-invariant if and only if $\varphi(\mathscr{F})\subset\mathscr{F}\otimes\Omega_{X}^{1}$. Now we can introduce the concept of slope stability. Recall that, given a torsion-free coherent sheaf $\mathscr{E}$ on an $n$-dimensional projective manifold $X$, the slope $\mu(\mathscr{E})$ of $\mathscr{E}$ with respect to $\omega$ is defined to be

$$\mu(\mathscr{E})=\frac{\deg_{\omega}(\mathscr{E})}{\mathrm{rank}\;\mathscr{E}}=\frac{c_{1}(\mathscr{E})\cdot\omega^{n-1}}{\mathrm{rank}\;\mathscr{E}}.$$

We will now focus on Higgs bundles, equivalently locally free Higgs sheaves, and will return to the general notion of Higgs sheaves in § 2.4, where such sheaves are constructed from spectral varieties defined by spectral data.

Let $E$ be a complex smooth vector bundle over $X$. We write $\Omega^{p,q}(E)$ for the complex vector space of $E$-valued $(p,q)$-forms on $X$. In the sequel of this paper, we will naturally identify the holomorphic structures on $E$ with the $\bar{\partial}$-operators $\bar{\partial}_{E}\colon\Omega^{p,q}(E)\to\Omega^{p,q+1}(E)$ satisfying the integrability condition $\bar{\partial}_{E}^{2}=0$. We denote by $\mathscr{E}:=(E,\bar{\partial}_{E})$ the holomorphic vector bundle with the holomorphic structure defined by $\bar{\partial}_{E}$ if there is no confusion.

Let $g\in\mathrm{Aut}(E)$. Then $g$ acts on the Higgs bundles $\mathscr{E}=((E,\bar{\partial}_{E}),\varphi)$ by $g\cdot(\bar{\partial}_{E},\varphi)=(g^{-1}\circ\bar{\partial}_{E}\circ g,g^{-1}\circ\Phi\circ g).$ We define the moduli stack of polystable Higgs bundles of rank $r$ as

(1)

$$\begin{split}\mathcal{M}^{{\rm stack},r}_{\mathrm{Higgs}}\coloneqq\{(\mathscr{E},\varphi)|(\mathscr{E},\varphi)\;\mathrm{polystable}\}/\mathrm{Aut}(E).\end{split}$$

A complex smooth vector bundle $E$ is said to be topologically trivial if all the Chern classes of $E$ in $H^{*}(X;\mathbb{Q})$ vanish. Let $[(\mathscr{E},\varphi)]$ be the equivalence class of $(\mathscr{E},\varphi)$ in the orbit of $\mathrm{Aut}(E)$. Under S-equivalence of $\mathrm{Aut}(E)$ action, a semi-stable topologically trivial Higgs bundle $[(\mathscr{E},\varphi)]$ is polystable. Following [Sim94b, Proposition 6.6], we define the Dolbeault moduli space $\mathcal{M}_{\mathrm{Dol}}^{r}$ as the moduli space parametrizing topologically trivial polystable Higgs bundles on $X$, which is a quasiprojective variety. It follows from [Sim88, Proposition 3.4] that a polystable Higgs bundle $(\mathscr{E},\varphi)$ is topologically trivial if and only if $c_{1}(\mathscr{E})\cdot\omega^{n-1}=0$ and $c_{2}(\mathscr{E})\cdot\omega^{n-2}=0$.

Let $X$ be a projective variety. The $\mathrm{GL}_{r}(\mathbb{C})$ character variety $\mathfrak{R}^{\mathrm{GL}_{r}(\mathbb{C})}$ of $X$ is defined to be the set of conjugacy classes of reductive representations of the fundamental group given by

(2)

$$\mathfrak{R}^{\mathrm{GL}_{r}(\mathbb{C})}\coloneqq\{\rho\colon\pi_{1}(X)\to\mathrm{GL}_{r}(\mathbb{C})\mid\rho\;\mathrm{reductive}\}/\sim.$$

The Hitchin morphism is a useful tool to study the moduli space of Higgs bundles. In this subsection, we will introduce the Hitchin morphism for projective manifolds, following [Hit87a, Hit87b, Sim94a]. Let $X$ be a projective manifold. Then the Hitchin base of $X$ with rank $r$ is defined to be

(4)

$$\mathcal{A}^{r}_{X}\coloneqq\bigoplus_{i=1}^{r}H^{0}(X,\mathrm{Sym}^{i}\Omega_{X}^{1}).$$

The Hitchin morphism for the moduli stack of Higgs bundle is defined as follows:

(5)

$$\begin{split}\mathsf{h}_{X}:\mathcal{M}_{\mathrm{Higgs}}^{\textup{stack},r}\to\mathcal{A}^{r}_{X},\quad[(\mathscr{E},\varphi)]\mapsto(\mathrm{Tr}(\varphi),\mathrm{Tr}(\varphi^{2})\cdots,\mathrm{Tr}(\varphi^{r})).\end{split}$$

We briefly recall the definition of the spectral base, which was introduced by T. Chen and B. Ngô in [CN20].

Let $V$ be a complex vector space of dimension $n$ and let $\mathrm{Chow}^{r}(V)$ be the Chow variety of zero cycles of length $r$ on $V$. By [CN20, Theorem 4.1], the following natural map

$$\mathrm{Chow}^{r}(V)\rightarrow V\times\mathrm{Sym}^{2}V\times\dots\times\mathrm{Sym}^{r}V,\quad[v_{1},\dots,v_{r}]\mapsto(\sigma_{1},\sigma_{2},\dots,\sigma_{r}),$$

is a closed embedding and thus it induces the following closed embedding

(6)

$$\mathrm{Chow}^{r}(\mathrm{Tot}(\Omega_{X}^{1})/X)\hookrightarrow\mathrm{Tot}(\Omega_{X}^{1})\times_{X}\mathrm{Tot}(\mathrm{Sym}^{2}\Omega_{X}^{1})\times_{X}\dots\times_{X}\mathrm{Tot}(\mathrm{Sym}^{r}\Omega_{X}^{1}),$$

where $\mathrm{Tot}(\bullet)$ denotes the total space of the corresponding vector bundle and the space $\mathrm{Chow}^{r}(\mathrm{Tot}(\Omega_{X}^{1})/X)$ is the relative Chow space of zero cycles of length $r$. In particular, under this closed embedding, the spectral base $\mathcal{S}^{r}_{X}$ can be identified with the space of sections $\sigma:X\rightarrow\mathrm{Chow}^{r}(\mathrm{Tot}(\Omega_{X}^{1})/X)$ and so $\mathcal{S}^{r}_{X}$ is a closed subset of $\mathcal{A}^{r}_{X}$.

The following observation shows that it suffices to check the condition in Definition 2.5 over general points to see whether an element $\textbf{s}\in\mathcal{A}^{r}_{X}$ is a spectral datum.

In [CN20], T. Chen and B. Ngô conjectured that $\mathsf{sd}_{X}$ is surjective. This conjecture has been confirmed in [CN20] and [SS24] for smooth projective surfaces, in [HL23] for rank two case and studied in [BKU23] for abelian variety. However, in general the moduli stack $\mathcal{M}_{\mathrm{Higgs}}^{{\rm stack},r}$ may be much larger than $\mathcal{M}_{\mathrm{Dol}}^{r}$. We are in particular interested in the image of the restriction of the spectral morphism to the Dolbeault moduli space, which leads to the following definition.

Clearly we have the natural inclusions $\mathcal{S}^{r}_{X;\mathrm{Dol}}\subset\mathcal{S}^{r}_{X}\subset\mathcal{A}^{r}_{X}$ and both of them are strict in the general case (see [HL23, Example 3.4]). Moreover, since by Theorem 2.4 the restriction $\mathsf{h}_{X}|_{\mathcal{M}_{\mathrm{Dol}}^{r}}$ is proper, the Dolbeault spectral base $\mathcal{S}^{r}_{X;\mathrm{Dol}}$ is a closed subset of $\mathcal{S}^{r}_{X}$.

Let $p:\mathrm{Tot}(\Omega_{X}^{1})\rightarrow X$ be the natural projection. Given a spectral datum $\textbf{s}=(s_{1},\dots,s_{r})\in\mathcal{S}^{r}_{X}$, the spectral variety $X_{\textbf{s}}$ corresponding to s is the closed subscheme of $\mathrm{Tot}(\Omega_{X}^{1})$ defined as follows

$$X_{\textbf{s}}\coloneqq\{\lambda^{r}-s_{1}\lambda^{r-1}+\dots+(-1)^{r-1}s_{r-1}\lambda+(-1)^{r}s_{r}=0\},$$

where $\lambda\in H^{0}(\mathrm{Tot}(\Omega_{X}^{1}),p^{*}\Omega_{X}^{1})$ is the Liouville form and for any $1\leq i\leq r$, the term $s_{i}\lambda^{r-i}$ is regarded as an element of $H^{0}(\mathrm{Tot}(\Omega_{X}^{1}),\mathrm{Sym}^{r}p^{*}\Omega_{X}^{1})$. In particular, the subscheme $X_{\textbf{s}}$ is locally defined by $\binom{n+r-1}{r-1}$ equations. To understand the spectral variety, we introduce the notion of multivalued holomorphic $1$-forms — see [CDY22, Definition 5.8].

Let $\widehat{X}_{\mathbf{s}}$ be the reduced scheme underlying $X_{\mathbf{s}}$; that is, $\widehat{X}_{\mathbf{s}}$ is the same topological space as $X_{\mathbf{s}}$, but with the reduced structure sheaf. Then the natural morphism $\pi:\widehat{X}_{\mathbf{s}}\rightarrow X$ is surjective and finite. In particular, there exists a dense Zariski open subset $X^{\circ}$ of $X$ such that $\widehat{X}^{\circ}_{\mathbf{s}}\coloneqq\pi^{-1}(X^{\circ})\rightarrow X_{\circ}$ is an unramified finite covering. Moreover, we can also write

$$\widehat{X}^{\circ}_{\mathbf{s}}=\bigcup_{k=1}^{m}\widehat{X}^{\circ}_{\mathbf{s},k}$$

for the decomposition of $\widehat{X}^{\circ}_{\mathbf{s}}$ into irreducible components. Since $\widehat{X}^{\circ}_{\mathbf{s}}\rightarrow X^{\circ}$ is unramified, the decomposition above is actually a disjoint union. On the other hand, one can easily see that each irreducible component $\widehat{X}^{\circ}_{\mathbf{s},k}$ defines a multivalued holomorphic $1$-form $[(\omega_{1}^{k},\dots,\omega^{k}_{r_{k}})]$ over $X^{\circ}$ whose local representatives have no multiple elements.

Now we can define the multiplicity of $\widehat{X}^{\circ}_{\mathbf{s},k}$ in the spectral variety $X_{\mathbf{s}}$. For any $k$ and $l$, we define the multiplicity $m(k,l)$ of the section $\omega^{k}_{l}$ to be its multiplicity as a root of the equation

$$\lambda^{r}-s_{1}\lambda^{n-1}+\dots+(-1)^{n-1}s_{n-1}\lambda+(-1)^{n}s_{n}=0.$$

We shall denote $m(k,l)$ by $m(k)$. Then we have $\sum_{k=1}^{m}m(k)r_{k}=r$. Let $\widehat{X}_{\mathbf{s},k}$ be the closure of $\widehat{X}^{\circ}_{\mathbf{s},k}$ in $\widehat{X}_{\mathbf{s}}$ and let $\hat{\pi}_{k}:\widehat{X}_{\mathbf{s},k}\rightarrow X$ be the natural morphism. We define

$$\hat{\mathscr{F}}_{k}\coloneqq\hat{\pi}_{k*}\mathscr{O}_{\hat{X}_{\mathbf{s},k}}.$$

Since $\hat{X}_{\mathbf{s},k}$ is integral, the structure sheaf $\mathscr{O}_{\hat{X}_{\mathbf{s},k}}$ is torsion-free. So $\hat{\mathscr{F}}_{k}$ is also torsion-free and it carries a natural Higgs field $\psi_{k}$ defined as follows:

(8)

$$\psi_{k}:\hat{\mathscr{F}}_{k}=\hat{\pi}_{k*}\mathscr{O}_{\widehat{X}_{\mathbf{s},k}}\xrightarrow{\times\lambda}\hat{\pi}_{k*}\left(\mathscr{O}_{\widehat{X}_{\mathbf{s},k}}\otimes\hat{\pi}_{k}^{*}\Omega_{X}^{1}\right)=\hat{\mathscr{F}}_{k}\otimes\Omega_{X}^{1}.$$

Recall that a coherent sheaf $\mathscr{F}$ over a complex manifold is said to be a normal sheaf if Hartogs’ extension across subvarieties of codimension $\geq 2$ holds, and $\mathscr{F}$ is said to be a reflexive sheaf if $\mathscr{F}^{**}=\mathscr{F}$. By [OSS11, Chapter 2, Lemma 1.1.12], a coherent sheaf on a complex manifold is reflexive if and only if it is normal and torsion-free.

Since $X$ is smooth and $\mathscr{E}_{k}$ is reflexive, there exists a dense Zariski open subset $X^{\circ\circ}$ of $X$ such that $X\setminus X^{\circ\circ}$ is of codimension $\geq 3$ in $X$ and such that the restriction $\mathscr{E}_{k}|_{X^{\circ\circ}}$ is locally free ([OSS11, Chapter 2, Lemma 1.1.10]). Denote by $\varphi_{k}^{\circ\circ}$ the restriction $\varphi_{k}|_{X^{\circ\circ}}$ and let

$$(s_{1k}^{\circ\circ},\cdots,s_{r_{k}k}^{\circ\circ})\coloneqq\mathsf{h}_{X^{\circ\circ}}\big{(}[\mathscr{E}_{k}|_{X^{\circ\circ}},\varphi_{k}^{\circ\circ}]\big{)}\in\bigoplus_{i=1}^{r_{k}}H^{0}(X^{\circ\circ},\mathrm{Sym}^{i}\Omega_{X^{\circ\circ}}^{1}).$$

Since $\mathrm{codim}(X\setminus X^{\circ\circ})\geq 3$ and $\mathrm{Sym}^{i}\Omega_{X}^{1}$ is locally free, the sections $s_{ik}^{\circ\circ}$ extend to sections $s_{ik}\in H^{0}(X,\mathrm{Sym}^{i}\Omega_{X}^{1})$. Let

$$P_{k}(T)\coloneqq T^{r_{k}}-s_{1k}T^{r_{k}-1}+\dots+(-1)^{r_{k}}s_{r_{k}k}$$

be the corresponding characteristic polynomial. Over the dense Zariski open subset $X^{\circ}$, one can easily derive the following equality of polynomials

(9)

$$P(T)\coloneqq T^{r}-s_{1}T^{r-1}+\cdots+(-1)^{k}s_{k}=\prod_{k=1}^{m}P_{k}(T)^{m(k)}.$$

Then it follows that the equality above actually holds over the whole $X$ by comparing the coefficients. Now let $X_{\mathbf{s},k}\subset\mathrm{Tot}(\Omega_{X}^{1})$ be the subscheme defined by $P_{k}(\lambda)=0$. Then clearly we have $X_{\mathbf{s},k}\cap p^{-1}(X^{\circ})=\widehat{X}^{\circ}_{\mathbf{s},k}$ and we may also write the equality (9) as an equation on cycles in the form

(10)

$$[X_{\mathbf{s}}]=\sum_{k=1}^{m}m(k)[X_{\mathbf{s},k}],$$

where for a pure-dimensional complex subspace $A$ of $X$, $[A]$ denotes the cycle in the Chow space Chow$(X)$ associated to $A$.

The Hitchin morphism (5) and the spectral morphism (7) can be directly extended to Higgs sheaves and one can immediately derive the following result from our argument above.

Let $\mathscr{L}$ be a holomorphic line bundle over a complex manifold $X$. We briefly recall the definition of a (singular) Hermitian metric on $\mathscr{L}$.

The curvature current of $\mathscr{L}$ is given formally by the closed $(1,1)$-current $\frac{\sqrt{-1}}{2\pi}\Theta_{\mathscr{L},h}=dd^{c}\varphi$ on $U$. The assumption $\varphi\in L^{1}_{\textup{loc}}$ guarantees that $\Theta_{\mathscr{L},h}$ exists in the sense of distribution theory. Moreover, for the curvature current for is globally defined over $X$ and independent of the choice of trivialisations, and its de Rham cohomology class is the image of the first Chern class $c_{1}(\mathscr{L})\in H^{2}(X,\mathbb{Z})$ in $H^{2}_{\textup{dR}}(X,\mathbb{R})$. If we assume in addition that $\varphi\in\mathcal{C}^{\infty}(U,\mathbb{R})$, then $h$ is the usual smooth Hermitian metric on $\mathscr{L}$.

Let $\mathscr{E}$ be a holomorphic vector bundle over a complex manifold $X$. Let $\mathbb{P}(\mathscr{E})$ be the projectivisation in the geometric sense, i.e., $\mathbb{P}(\mathscr{E})$ parametrises the one-dimensional linear subspaces contained in the fibres of $\mathscr{E}$. Let $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(-1)\subset\pi^{*}\mathscr{E}$ be the dual tautological line bundle over $\mathbb{P}(\mathscr{E})$, where $\pi:\mathbb{P}(\mathscr{E})\rightarrow X$ is the natural projection. Given a (singular) Hermitian metric $h$ over $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(-1)$, we can define a pseudo-metric $\bar{h}$ over $\mathscr{E}$ as in the following. For any $v\in\mathscr{E}_{x}\setminus\{0\}$, we define

$$\|v\|_{\bar{h}}:=\|v\|_{h},$$

where on the right-hand side we regard $v$ as the corresponding point in the fibre of the natural projection $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(-1)\rightarrow\mathbb{P}(\mathscr{E})$ over $[v]$. Such a metric $\bar{h}$ is called a (complex) Finsler pseudo-metric and we call it a (complex) Finsler metric if the metric $h$ is a smooth Hermitian metric.

We collect some basic definitions and facts about bounded symmetric domains and we refer the interested reader to [Mok89] for more details. Let $\Omega\Subset\mathbb{C}^{n}$ be a bounded domain in a complex Euclidean space. We say that $\Omega$ is a bounded symmetric domain if and only if at each $x\in\Omega$, there exists a biholomorphism $\sigma_{x}:\Omega\rightarrow\Omega$ such that $\sigma_{x}^{2}=\mathrm{id}$ and $x$ is an isolated fixed point of $\sigma_{x}$. In this case, the Bergman metric $ds_{\Omega}^{2}$ with Kähler form $\omega$ on $\Omega$ is Kähler–Einstein and $(\Omega,\omega)$ is a Hermitian symmetric space of the non-compact type. The rank of $\Omega$ is defined to be the rank of $(\Omega,ds_{\Omega}^{2})$ as a Riemannian symmetric manifold. We say that the bounded symmetric domain $\Omega$ is irreducible if and only if $(\Omega,ds^{2}_{\Omega})$ is an irreducible Riemannian symmetric manifold. Denote by $\mathrm{Aut}(\Omega)$ the group of biholomorphic self-mappings on $\Omega$. Write $G$ for the identity component $\mathrm{Aut}_{o}(\Omega)$ and let $K\subset G$ be the isotropy subgroup at a point $o\in\Omega$, so that $\Omega=G/K$ as a homogeneous space.

Now we introduce the minimal characteristic bundle — see [Mok89, Chapter 6,§ 1] and [Mok02]. Let $\Omega$ be an irreducible bounded symmetric domain. Then we can identify $\Omega$ as a subdomain of its compact dual $M$ by the Borel embedding ([Mok89, Chapter 3, § 3]). Let $G^{\mathbb{C}}$ be the automorphism group of $M$ and $P\subset G$ be the isotropy subgroup at $o$. Then $G^{\mathbb{C}}\supset G$ is a complexification of $G$. Consider the action of $P$ on $\mathbb{P}(T_{o}M)$. There are exactly $r$ orbits $\mathcal{O}_{k}\subset\mathbb{P}(T_{o}M)=\mathbb{P}(T_{o}\Omega)$, $1\leq k\leq r$, such that the topological closures $\bar{\mathcal{O}}_{k}$ form an ascending chain of subvarieties of $\mathbb{P}(T_{o}M)$ with $\bar{\mathcal{O}}_{r}=\mathbb{P}(T_{o}M)$. In particular, the variety $\mathcal{O}_{1}$ is the unique closed orbit, which is thus a homogeneous projective submanifold of $\mathbb{P}(T_{o}M)$. Moreover, the submanifold $\mathcal{O}_{1}\subset\mathbb{P}(T_{o}M)$ is nothing other than the variety of minimal rational tangents (VMRT) of $M$ at $o$, i.e., the variety of tangent directions at $o$ of projective lines on $M$ passing through $o$ with respect to the first canonical projective embedding of $M$. In particular, the subvariety $\mathcal{O}_{1}\subset\mathbb{P}(T_{o}M)$ is linearly non-degenerate, i.e., not contained in a hyperplane. The $G$-orbit $\mathcal{S}(\Omega)$ of a point $0\not=[\eta]\in\mathcal{O}_{1}$ is a holomorphic bundle of homogeneous projective manifolds over $\Omega$, which is called the minimal characteristic bundle of $\Omega$.

Let $\Omega$ be an arbitrary bounded symmetric domain, and write $\Omega=\Omega_{1}\times\cdots\times\Omega_{m}$ for its decomposition into the Cartesian product of its irreducible factors. Write $T\Omega=T_{1}\oplus\cdots\oplus T_{m}$ for the corresponding direct sum decomposition of the holomorphic tangent bundle. Let us denote by $\mathcal{S}^{i}(\Omega)\subset\mathbb{P}(T{\Omega})$ the holomorphic bundle over $\Omega$ obtained from the natural embedding of $\mathcal{S}(\Omega_{i})\subset\mathbb{P}(T\Omega_{i})$ into the projective subbundle $\mathbb{P}(T_{i})\subset\mathbb{P}(T\Omega)$. Let $X=\Omega/\Gamma$ be the quotient space by a torsion-free irreducible cocompact lattice. Then the Bergman metric $ds^{2}_{\Omega}$ on $\Omega$ descends to a quotient metric $g$ on $X$, which is again a Kähler-Einstein metric. We have the following Finsler metric rigidity theorem on $X$ proved by the last author in [Mok04].