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Extensions of Formal Hodge Structures

Nicola Mazzari

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.

Extensions of Formal Hodge Structures

Abstract

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

Paper Structure

This paper contains 10 sections, 18 theorems, 56 equations.

Table of Contents

  1. Formal Hodge Structures
  2. Sub-categories of ${\sf FHS}_n$
  3. Adjunctions
  4. Different levels
  5. Extensions in ${\sf FHS}_n$
  6. Basic facts
  7. Formal Carlson theory
  8. Sharp Cohomology
  9. The generalized Albanese of Esnault-Srinivas-Viehweg
  10. The generalized Albanese of Faltings and Wüstholz

Key Result

Proposition 1.7

i) The category ${\sf FHS}_n$ is an abelian category. ii) The forgetful functor $(H,V)\mapsto H$ (resp. $(H,V)\mapsto V$) is an exact functor with values in the category of formal groups (resp. the category ${\sf Vec}_n$). iii) There exists a full and thick embedding ${\sf MHS}_l(0)\to {\sf FHS}_l(0

Theorems & Definitions (63)

  • Definition 1.1: level $=0$
  • Definition 1.2: level $\le n$
  • Remark 1.3
  • Example 1.4: Sharp cohomology of a curve
  • Remark 1.5: Twisted fhs
  • Example 1.6: Level $\le 1$
  • Proposition 1.7: Properties of FHS
  • proof
  • Lemma 1.8
  • proof
  • ...and 53 more