Related papers: A Note on Hodge theoretic anabelian geometry
We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are…
We study the various arithmetic and geometric Frobenius morphisms on the moduli stack of principal bundles over a smooth projective algebraic curve and determine explicitly their actions on the $\ell-$adic cohomology of the moduli stack in…
We use Hodge-theoretic methods to (i) explain number-theoretic identities of a type recently considered by Guillera and Zudilin, (ii) describe the Frobenius dual of Abel-Jacobi period functions, and (iii) offer a new proof of Golyshev's…
A theorem of Grothendieck asserts that over a perfect field k of cohomological dimension one, all non-abelian H^2-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of…
It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad…
A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes…
Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field $K$ of characteristic zero. We use this notion of hyperbolicity to show the following…
This is a survey of results and conjectures on mirror symmetry phenomena in the non-Abelian Hodge theory of a curve. We start with the conjecture of Hausel-Thaddeus which claims that certain Hodge numbers of moduli spaces of flat SL(n,C)…
We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the…
In this paper we investigate the arithmetic aspects of the theory of $\mathcal{E}_K^\dagger$-valued rigid cohomology introduced and studied in [11,12]. In particular we show that these cohomology groups have compatible connections and…
In anabelian geometry, we consider to what extent the \'{e}tale or tame fundamental groups of schemes reflect geometric properties of the schemes. Although there are many known results (mainly for smooth curves) in this area, general…
Hodge classes on the moduli space of admissible covers with monodromy group G are associated to irreducible representations of G. We evaluate all linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy in terms…
This paper studies the Galois action on a special lattice of geometric origin, which is related to mod-$\ell$ abelian-by-central quotients of geometric fundamental groups of varieties. As a consequence, we formulate and prove the mod-$\ell$…
We introduce a new approach to the geometric Bombieri--Lang conjecture for hyperbolic varieties in characteristic 0. The main idea is to construct an entire curve on a special fiber of a variety over a complex function field from an…
The goal of this paper is to first define a Hodge theoretic fundamental group for smooth connected complex algebraic varieties and then prove and study a right exact sequence of Hodge theoretic fundamental groups associated to a smooth…
Let $X$ be a proper, smooth, and geometrically connected curve of genus $g(X)\ge 1$ over a $p$-adic local field. We prove that there exists an effectively computable open affine subscheme $U\subset X$ with the property that $period (X)=1$,…
The classical Rees construction (of common use in commutative algebra and Hodge theory) interpolates between filtrations, viewed as ${\mathbb G}_m$-equivariant vector bundles on the affine line, and their associated gradings. Various…
This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian…
This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of…
Hodge theory associates to a smooth projective variety over $\mathbb{C}$ a piece of linear algebra information, called a $\mathbb{Q}$-Hodge structure. Conversely, it is a natural question which abstract $\mathbb{Q}$-Hodge structures arise…