Impartido por Eva Elduque (Universidad Autónoma de Madrid), del 10 al 12 de junio

24 may 2024 - 11:43 CET

Conference Symplectic and Algebraic Geometryof real and p-adic systems

Madrid, June 10–12, 2024

Minicourse:

Deligne’s theory of mixed Hodge systems

Eva Elduque

Universidad Autónoma de Madrid

The Hodge structure on the (co)homology of a smooth compact complex algebraic va- riety is a collection of linear algebraic data that provides many obstructions

and insights about its topology. In the seventies, Deligne showed that the theory can be extended to both non-smooth and non-compact varieties by expanding

the notion of Hodge structure to that of a mixed Hodge structure, enabling these tools to be used in a much broader setting. In this course, we will give an introduction

to Deligne’s theory of mixed Hodge complexes of sheaves in the setting of smooth non-compact varieties, as well as talk about recent develop- ments using these techniques.

TUESDAY June 11th, 10:00-11:20, Aula Miguel de Guzmán

Lecture 1: Hodge structures. We will review the construction of the Hodge structure on the cohomology of compact Kähler manifolds, giving a

sheaf-theoretic interpretation. We will also define the notion of mixed Hodge structure.

TUESDAY June 11th, 14:30-15:50, Aula Miguel de Guzmán

Lecture 2: Mixed Hodge structures and mixed Hodge complexes of sheaves. We will define the notion of a mixed Hodge complex of sheaves, and recall

Deligne’s construction of the mixed Hodge structure on the cohomology of smooth algebraic varieties.

WEDNESDAY June 12th, 11:50-13:10, Aula Miguel de Guzmán

Lecture 3: New developments using Deligne’s techniques. We will talk about a new Hodge theory for abelian covers of smooth algebraic varieties.

These covers are complex analytic manifolds, but not algebraic in general. This will be based on joint work with C. Geske, M. Herradón Cueto, L. Maxim and B. Wang.