Extensions of Formal Hodge Structures

TL;DR

This paper extends the concept of formal Hodge structures to higher levels, exploring their properties and applying them to describe generalized Albanese varieties for complex varieties that are not necessarily smooth.

Contribution

It introduces the category of formal Hodge structures of level ≤ n and generalizes the classical Albanese variety formula to broader classes of complex varieties.

Findings

Defined the category ${ m FHS}_n$ for higher levels.

Connected ${ m Ext}^1$ groups in ${ m FHS}_n$ to generalized Albanese varieties.

Extended classical results to proper, non-smooth complex varieties.

Abstract

We define and study the properties of the category ${\sf FHS}_n$ of formal Hodge structure of level $\le n$ following the ideas of L. Barbieri-Viale who discussed the case of level $\le 1$. As an application we describe the generalized Albanese variety of Esnault, Srinivas and Viehweg via the group $\Ext^1$ in ${\sf FHS}_n$. This formula generalizes the classical one to the case of proper but non necessarily smooth complex varieties.