On the equidistribution of some Hodge loci

We prove the equidistribution of the Hodge locus for certain non-isotrivial, polarized variations of Hodge structure of weight 2 with {h^{2,0}=1} over complex, quasi-projective curves. Given some norm condition, we also give an asymptotic on the growth of the Hodge locus. In particular, this implies the equidistribution of elliptic fibrations in quasi-polarized, non-isotrivial families of K3 surfaces.

On Motivic Realizations of the Canonical Hermitian Variations of Hodge Structure of Calabi–Yau Type over type Domains

Let${\mathcal{D}}$be the irreducible Hermitian symmetric domain of type$D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian variation of real Hodge structure${\mathcal{V}}_{\mathbb{R}}$of Calabi–Yau type over ${\mathcal{D}}$. This short note concerns the problem of giving motivic realizations for ${\mathcal{V}}_{\mathbb{R}}$. Namely, we specify a descent of${\mathcal{V}}_{\mathbb{R}}$from$\mathbb{R}$to$\mathbb{Q}$and ask whether the$\mathbb{Q}$-descent of${\mathcal{V}}_{\mathbb{R}}$can be realized as sub-variation of rational Hodge structure of those coming from families of algebraic varieties. When$n=2$, we give a motivic realization for ${\mathcal{V}}_{\mathbb{R}}$. When$n\geqslant 3$, we show that the unique irreducible factor of Calabi–Yau type in$\text{Sym}^{2}{\mathcal{V}}_{\mathbb{R}}$can be realized motivically.

Introduction to Variations of Hodge Structure

This chapter emphasizes the theory of abstract variations of Hodge structure (VHS) and, in particular, their asymptotic behavior. It first studies the basic correspondence between local systems, representations of the fundamental group, and bundles with a flat connection. The chapter then turns to analytic families of smooth projective varieties, the Kodaira–Spencer map, Griffiths' period map, and a discussion of its main properties: holomorphicity and horizontality. These properties motivate the notion of an abstract VHS. Next, the chapter defines the classifying spaces for polarized Hodge structures and studies some of their basic properties. Finally, the chapter deals with the asymptotics of a period mapping with particular attention to Schmid's orbit theorems.

Hodge Theory (MN-49)

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.

Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising From Mirror Symmetry and Middle Convolution

We collect evidence in support of a conjecture of Griffiths, Green, and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi–Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions 1 ≤ d ≤ 6 arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is G2) of the family of 6-folds, and the theory of boundary components of Mumford–Tate domains.

The double cover of cubic surfaces branched along their Hessian

We establish a relation between the Hodge structure of the double cover of a nonsingular cubic surface branched along its Hessian and the Hodge structure of the triple cover of P3 branched along the cubic surface. Then we introduce a method to study the infinitesimal variations of Hodge structure of the double cover of the cubic surface. Using these results, we compute the Néron–Severi lattices for the double cover of a generic cubic surface and the Fermat cubic surface.