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Donu Arapura

Publications and source records attributed to Donu Arapura.

At least 19 recordsLinked to original sources
arXiv2026

A Higher dimensional log Riemann--Hurwitz inequality and rigidity of covers

Let $\overline{Z}$ be a smooth projective variety with an simple normal crossing divisor $D$ such that $Ω_{\overline{Z}}^1(\log D)$ is nef. We prove that if $Y\subset Z:=\overline Z\setminus D$ is a smooth closed subvariety with nonzero Euler characteristic, and $P$ is a perverse sheaf on $Y$ with full support, then $χ(Y,P)>0$. This is a strict version of an inequality obtained in arXiv:2408.15788. Applying this to the trace-zero part of a finite direct image yields a logarithmic Riemann--Hurwitz inequality: if $Y$ has dimension $n$, any finite surjective morphism $f\colon X\to Y$ of degree $d$ with $X$ smooth satisfies $(-1)^nχ(X)\ge d\,(-1)^nχ(Y)$, the difference being an explicit sum of nonnegative intersection numbers. When $(-1)^nχ(Y)>0$ this forces any such $f$ with $χ(X)=χ(Y)$ to be an isomorphism. We verify the nef hypothesis for subvarieties of semiabelian varieties, and for $\overline{\mathscr M}_{g,n}$---the moduli of curves, obtaining in particular that every finite surjective self-morphism of a moduli space of curves with level structure is an isomorphism.

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arXiv2026

Differential Forms and Hodge Structures on Singular Varieties

We compare a couple of notions of differential form on singular complex algebraic varieties, and relate them to the outermost associated graded spaces of the Hodge filtration of ordinary and intersection cohomology. In particular, we introduce and study singularities, that we call quasi-rational, which are normal and such that for all p, the zeroth cohomology sheaf of the complex of Du Bois p-forms is isomorphic to the direct image of p-forms from a desingularization. We show that an isolated singularity is rational if and only if it is quasi-rational, Du Bois, and certain Hodge numbers of the local mixed Hodge structures vanish.

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arXiv2026

Failure of the invariant cycle theorem over $\mathbb Z$

We initiate a study of the local invariant cycle theorem with integral coefficients for 1-parameter semistable families of varieties. We show that it always holds for $H^1$, and it holds for $H^2$ if the general fiber has trivial Albanese variety. The latter generalizes results of Friedman, Griffiths, and Scattone on K3 surfaces and I-surfaces. We construct the first example of a semistable family which fails the local (and global) invariant cycle theorems with integral coefficients. The family has constant period map associated to $H^2$, and its smooth fibers are algebraic surfaces with $p_g=q=1$; in particular, they have non-trivial Albanese varieties. The surfaces in the family have maximal Picard rank and minimal discriminant, and they are closely related to Vinberg's most algebraic K3 surface. Our construction also generalizes the Shioda--Inose construction for rational double covers of K3 surfaces.

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arXiv2025

Nonnegativity of signed Euler characteristics of moduli of curves and abelian varieties

Given a perverse sheaf on the moduli stack of principally polarized abelian varieties or the moduli stack of smooth curves with n marked points over a field of characteristic zero, we prove that the (orbifold) Euler characteristic is nonnegative. For constant coefficients, this follows immediately from formulas of Harer-Zagier and Harder. Our proof is different and in the case of abelian varieties uses log Dubson-Kashiwara plus the fact that Hodge bundles are nef. For curves, we require an additional inequality established using Beilinson's gluing construction. The first main result is shown to be false in positive characteristic.

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arXiv2025

Euler characteristics of Kollár-hyperbolic varieties

Call a normal complex projective variety $X$ Kollár-hyperbolic if any nonconstant map from a smooth projective curve to $X$ induces a nontrivial homomorphism of étale fundamental groups. Examples include (a) smooth varieties with finite Albanese map, (b) normalizations of subvarieties of hermitian locally symmetric varieties of noncompact type, and (c) higher dimensional Kodaira fibrations. We conjecture that Kollár-hyperbolic varieties satisfy a vanishing theorem, which says roughly that if $P$ is perverse sheaf underlying a mixed Hodge module on such a variety then the limit of normalized dimensions of the cohomology groups of $P$ are zero in nonzero degrees, where the limit is taken over a suitable tower of étale covers. We call such varieties V-hyperbolic. V-hyperbolic varieties satisfy a Gromov type vanishing theorem for $L^2$ cohomology, the inequalities $(-1)^dχ(X) \ge 0$ and $(-1)^{d-p}χ(Ω_X^p)\ge 0$ in the smooth case, and more generally, an inequality for mixed Hodge modules conjectured under related assumptions by Maxim, Wang and the author. We prove that examples of type (a) and (c) listed above are V-hyperbolic.

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arXiv2025

Universal Chern classes on the moduli of bundles

The goal of this paper is to construct universal cohomology classes on the moduli space of stable bundles over a curve when it is not a fine moduli space, i.e. when the rank and degree are not coprime. More precisely, we show that certain Chern classes of the universal bundle on the product of the curve with the moduli stack of bundles lift to the product of the curve with the moduli space of stable bundles.

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arXiv2024

Equivariant Hodge modules and rational singularities

We define a notion of Hodge modules with rational singularities. A variety has rational singularities in the usual sense, if it is normal and the Hodge module related to intersection cohomology has rational singularities in the present sense. Our main result is a generalization of Boutot's theorem that if a reductive group acts on an affine variety with a stable point, and $H$ is an equivariant Hodge module with rational singularities, then the induced module on the GIT quotient also has rational singularities.

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arXiv2023

Perverse sheaves on varieties with large fundamental groups

We conjecture that any perverse sheaf on a compact aspherical Kähler manifold has non-negative Euler characteristic. This extends the Singer-Hopf conjecture in the Kähler setting. We verify the stronger conjecture when the manifold X has non-positive holomorphic bisectional curvature. We also show that the conjecture holds when X is projective and in possession of a faithful semi-simple rigid local system. The first result is proved by expressing the Euler characteristic as an intersection number involving the characteristic cycle, and then using the curvature conditions to deduce non-negativity. For the second result, we have that the local system underlies a complex variation of Hodge structure. We then deduce the desired inequality from the curvature properties of the image of the period map.

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arXiv2022

Hodge cycles and the Leray filtration

This is loosely a continuation of the author's previous paper arXiv:1802.09496. In the first part, given a fibered variety, we pull back the Leray filtration to the Chow group, and use this to give some criteria for the Hodge and Tate conjectures to hold for such varieties. In the second part, we show that the Hodge conjecture holds for a good desingularization of a self fibre product of a non-isotrivial elliptic surface under appropriate conditions. We also show that the Hodge and Tate conjectures hold for natural families of abelian varieties parameterized by certain Shimura curves. This uses Zucker's description of the mixed Hodge structure on the cohomology of a variation of Hodge structures on a curve, along with appropriate "vanishing" theorems.

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arXiv2022

Motivic sheaves revisited

In earlier work (arXiv:0801.0261), we gave a definition of an abelian category of motivic (constructible) sheaves over a base in characteristic zero using Nori's method. This category has Hodge and etale realizations, and is stable under inverse and direct images for projective and constant maps. The goal of this paper is to give a simpler definition of (a slight modification of) the category, and to give an easier proof of the direct image theorem. This paper relies on the previous article for some technical results, but is otherwise self contained.

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arXiv2022

Residues of connections and the Chevalley-Weil formula for curves

Given a finite group of automorphisms of a compact Riemann surface, the Chevalley-Weil formula computes the character valued Euler characteristic of an equivariant line bundle. The goal of this article is to give a proof by computing using residues of a Gauss-Manin connection.

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arXiv2020

When are braid groups of manifolds Kähler?

Sometime ago, we showed that a pure Artin braid group is not Kähler, i.e. it is not the fundamental group of a compact Kähler manifold. This used a result of Bressler, Ramachandran and the author that Kähler groups cannot be too "big". The goal here is to study the problem of Kählerness for other braid groups. The main result is that, with some trivial exceptions, the pure braid group of a Riemann surface with at least 2 strands is never Kähler. In some cases the proof uses the previous strategy, for others it plays off some homological properties of braid groups established beforehand against consequences of the Beauville-Catanese-Siu theorem. The braid group of a projective manifold of complex dimension 2 or more is shown to the fundamental group of a projective manifold, and hence Kähler.

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arXiv2018

Algebraic cycles on genus two modular fourfolds

We study universal families of stable genus two curves with level structure. Among other things, it is shown that the (1,1) part is spanned by divisor classes, and that there are no cycles of type (2,2) in the third cohomology of the first direct image. Using this, we deduce the Hodge and Tate conjectures hold for these varieties.

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arXiv2018

Kodaira vanishing for singular varieties revisited

We correct the proof and slightly strengthen a Kodaira-type vanishing theorem for singular varieties originally due to Jaffe and the first author. Specifically, we show that if $L$ is a nef and big line bundle on a projective variety of characteristic zero, the $i^{\text{th}}$ cohomology of $L^{-1}$ vanishes for $i$ in a range determined by the depth and dimension of the singular locus.

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arXiv2018

Vanishing theorems for parabolic Higgs bundles

This is a sequel to "Kodaira-Saito vanishing via Higgs bundles in positive characteristic" (arXiv:1611.09880). However, unlike the previous paper, all the arguments here are in characteristic zero. The main result is a Kodaira vanishing theorem for semistable parabolic Higgs bundles with trivial parabolic Chern classes. This implies a general semipositivity theorem. This also implies a Kodaira-Saito vanishing theorem for complex variations of Hodge structure.

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arXiv2017

Lefschetz decompositions for eigenforms on a Kähler manifold

We show that the eigenspaces of the Laplacian $Δ_k$ on $k$-forms on a compact Kähler manifold carry Hodge and Lefschetz decompositions. Among other consequences, we show that the positive part of the spectrum of $Δ_k$ lies in the spectrum of $Δ_{k+1}$.

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