Eva Elduque

Universidad Autónoma de Madrid

 

Conference Symplectic and Algebraic Geometry of real and p-adic systems

Madrid, June 10–12, 2024

 

 

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.