Extensions of Formal Hodge Structures

Abstract

We define and study the properties of the category of formal Hodge structure of level ≤n following the ideas of Barbieri-Viale who discussed the case of level ≤1. As an application, we describe the generalized Albanese variety of Esnault, Srinivas, and Viehweg via the group Ext¹ in . This formula generalizes the classical one to the case of proper but not necessarily smooth complex varieties.

Key Words:

• Albanese
• Hodge structures

2000 Mathematics Subject Classification:

• 14 F (14C30)

ACKNOWLEDGMENTS

The author would like to thank L. Barbieri-Viale for pointing his attention to this subject and for helpful discussions. The author also thanks A. Bertapelle for many useful comments and suggestions.

Notes

By thick we mean a subcategory closed under kernels, co-kernels, and extensions.

The superscript prp stands for “proper”. In fact, the sharp cohomology objects (3.1) of a proper variety have this property.

By injective resolution of a complex of sheaves A ^• we mean a quasi isomorphism A ^• → I ^•, where I ^• is a complex of injective objects.

It is possible to replace ℂ with a field k of characteristic zero. In that case, we must assume that there exists a k rational point in order to have FW(Z) defined over k .

Communicated by C. Pedrini.