An Introduction to Hodge Structures

TL;DR

This paper introduces Hodge structures, explores their properties, and discusses their applications in complex geometry, including period maps, variations, degenerations, and connections to the Gross-Siebert program.

Contribution

It provides a comprehensive overview of Hodge theory, including recent developments and their relevance to degenerations and mirror symmetry.

Findings

Description of Hodge and Lefschetz decompositions

Analysis of the period map and its properties

Discussion of mixed Hodge structures and degenerations

Abstract

We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid's limiting mixed Hodge structure and Steenbrink's work in the geometric setting. Finally, we give an outlook about Hodge theory in the Gross-Siebert program.