We study the Hitchin morphism for higher dimensional varieties and show that, for a certain class of varieties which we call r-small, the set-theoretic image of the Hitchin morphism from the Dolbeault moduli space coincides with the spectral base. In other words, a stronger version of the conjecture of Chen and Ngô holds for this class of varieties, which includes K-trivial varieties. As part of the proof, we slightly modify the construction of spectral covers to obtain normal spectral covers.
In this paper, we explore the structure of the Hitchin morphism for higher dimensional varieties. We show that the Hitchin morphism factors through a closed subscheme of the Hitchin base, which is in general a non-linear subspace of lower dimension. We conjecture that the resulting morphism, which we call the spectral data morphism, is surjective. In the course of the proof, we establish connections between the Hitchin morphisms for higher dimensional varieties, the invariant theory of the commuting schemes, and Weyl's polarization theorem. We use the factorization of the Hitchin morphism to c…
The Hitchin morphism is a map from the moduli space of Higgs bundles $\mathscr{M}_X$ to the Hitchin base $\mathscr{B}_X$, where $X$ is a smooth projective variety. When $X$ has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme $\mathscr{A}_X$ of $\mathscr{B}_X$, which is called the space of spectral data. They proved that the Hitchin morphism factors through $\mathscr{A}_X$ and conjectured that $\mathscr{A}_X$ is the image of the Hitchin morphism. We prove that when $X$ is a smooth projective surface, this conjecture is true fo…
We initiate and develop a framework to handle the specialization morphism as a filtered morphism for the perverse, and for the perverse Leray filtration, on the cohomology with constructible coefficients of varieties and morphisms parameterized by a curve. As an application, we use this framework to carry out a detailed study of filtered specialization for the Hitchin morphisms associated with the compactification of Dolbeault moduli spaces in [de-2018].
On the image of Hitchin morphism for some classical groups on algebraic surfaces
Beijing Institute of Mathematical Sciences and Applications（北京数学与应用科学研究所）；Massachusetts Institute of Technology, IAiFi Institute（麻省理工学院 IAiFi 研究所）；National Research University, Higher School of Economics, Russian Federation, Laboratory of Mirror Symmetry（俄罗斯联邦国立研究型大学高等经济大学镜像对称实验室）
We describe certain Lagrangian fibrations by Prym varieties on holomorphic symplectic varieties that deform to compactifications of Sp-Hitchin systems.
We prove that the perverse Leray filtration for the Hitchin morphism is locally constant in families, thus providing some evidence towards the validity of the $P=W$ conjecture due to de Cataldo, Hausel and Migliorini in non Abelian Hodge theory.
The aim of this paper is to construct certain closed embeddings of Grassmannian varieties, using tensor operations on vector bundles. These embeddings generalize Segre and Plücker morphisms.
Hitchin in [Duke Math. J. 54 (1), 91-114 (1987)] introduced a proper morphism from the moduli space of stable $G$-Higgs bundles ($G=\mathrm{GL}(n,\mathbb{C}),\mathrm{Sp}(2m,\mathbb{C})$ and $\mathrm{SO}(n,\mathbb{C})$) over a curve to a vector space of invariant polynomials and he described the generic fibers of that morphism. In this paper, we first describe the generic Hitchin fibers for the moduli space of stable parabolic projective symplectic/orthogonal Higgs bundles without fixing the determinant. We also describe the generic fibers when the determinant is trivial.
Local obstructions to the surjectivity of spectral morphisms
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Max Planck Institute for Mathematics（马普数学研究所）
For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Picard rank one, and show that they are connected and contractible.
The goal of this paper is to introduce a new constructive geometric proof of the affine version of Chevalley's Theorem. This proof is algorithmic and a verbatim implementation resulted in an efficient code for computing the constructible image of rational maps between affine varieties. Our approach extends the known descriptions of uniform matrix product states to $\operatorname{uMPS}(2,2,5)$
Let $f \colon X \to A$ be a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). We show that the sheaves $f_* ω_X^{\otimes m}$ become globally generated after pullback by an isogeny. We use this to deduce a decomposition theorem for these sheaves when $m \ge 2$, analogous to that obtained by Chen-Jiang when $m = 1$. This is in turn applied to effective results for pluricanonical linear series on irregular varieties with canonical singularities.
Let $f$ be a morphism from a klt pair $(X, Δ)$ to an abelian variety $A$, $m\geq1$ a rational number and $D$ a Cartier divisor on $X$ such that $D\sim_{\mathbb Q}m(K_X+Δ)$. We prove that the sheaf $f_*\mathcal{O}_X(D)$ becomes globally generated after pullback by an isogeny and has the Chen-Jiang decomposition, along with some related results. These are applied to some effective results for $\mathcal{O}_X(D)$ when $X$ is irregular.
Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.
Let $X$ be a projective variety (possibly singular) over an algebraically closed field of any characteristic and $\mathcal{F}$ be a coherent sheaf. In this article, we define the determinant of $\mathcal{F}$ such that it agrees with the classical definition of determinant in the case when $X$ is non-singular. We study how the Hilbert polynomial of the determinant varies in families of singular varieties. Consider a singular family such that every fiber is a normal, projective variety. Unlike in the case when the family is smooth, the Hilbert polynomial of the determinant does not remain consta…
Over a field of positive characteristic, a semisimple algebraic group $G$ may have some nonreduced parabolic subgroup $P$. In this paper, we study the Schubert and Bott-Samelson-Demazure-Hansen (BSDH) varieties of $G/P$, with $P$ nonreduced, when the base field is perfect. It is shown that in general the Schubert and BSDH varieties of such a $G/P$ are not normal, and the projection of the BSDH variety onto the Schubert variety has nonreduced fibers at closed points. When the base field is finite, the generalized convolution morphisms between BSDH varieties are also studied. It is shown that th…
We prove that every smooth subelliptic variety admits a surjective morphism from an affine space. This result gives partial answers to the questions of Arzhantsev and Forstnerič. As an application, we characterize open images of morphisms between affine spaces. We also obtain the jet interpolation theorem for morphisms from zero-dimensional subschemes of affine varieties to smooth subelliptic varieties.
We describe spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications we study irreducible components of singular Hitchin fibre…
We prove a conjecture by Kawaguchi-Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties.
We study finite morphisms of varieties and the link between their top multiplicity loci under certain assumptions. More precisely, we focus on how to determine that link in terms of the spaces of arcs of the varieties.
Let X and Y be nonsingular real algebraic varieties, dimX>dimY-1. Assume that the variety Y is malleable, compact and connected. Our main result implies that each regular map from X to Y is homotopic to a surjective regular map. The class of malleable varieties includes all homogeneous spaces for linear real algebraic groups.
We calculate the Picard group of generic (very general) spectral varieties living in the total space of a very ample line bundle over an algebraically closed field $k$ of odd characteristics or characteristic 0. We follow the strategy of Ravindra and Srinivas [RS06,RS09] via formal Picard groups. As an application, we calculate the generic fibers of Hitchin systems over a smooth quintic surface of Picard number 1.
Let $K$ be a field of characteristic $0$ and let $G$ and $H$ be connected commutative algebraic groups over $K$. Let $\text{Mor}_0(G,H)$ denote the set of morphisms of algebraic varieties $G \to H$ that map the neutral element to the neutral element. We construct a natural retraction from $\text{Mor}_0(G,H)$ to $\text{Hom}(G,H)$ (for arbitrary $G$ and $H$) which commutes with the composition and addition of morphisms. In particular, if $G$ and $H$ are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If $G$ has no non-trivial unipotent group as a direct factor, w…
