Structures de Hodge mixtes et fibr\'es sur le plan projectif complexe

TL;DR

This paper geometrizes mixed Hodge structures by associating equivariant vector bundles on the complex projective plane, establishing an equivalence of categories, and introducing a new invariant related to the second Chern class.

Contribution

It constructs a categorical equivalence between mixed Hodge structures and semistable equivariant vector bundles, and defines a new invariant generalizing the R-split level.

Findings

Establishes a dictionary linking mixed Hodge structures to equivariant vector bundles.

Defines the R-split level using the second Chern class of Rees bundles.

Shows semicontinuity of Hodge-type integers in families of structures.

Abstract

The purpose of this work is to geometrize the notion of mixed Hodge structure. Therefore, we associate equivariant vector bundles on the projective plane to trifiltered vector spaces. Making this Rees construction with filtrations arising from mixed Hodge structures gives an equivalence of categories between the category of mixed Hodge structures and a category of equivariant vector bundles on ${\bf P}^{2}_{\bf C}$ that are semistable for a semistability condition with respect to a divisor. Using this dictionary we can recover geometrically some properties of mixed Hodge structures like the abelian caracter of the category of mixed Hodge structures or the fact that higher extensions vanish. Next, looking at the second Chern class of the Rees bundles, we define the ${\bf R}$-split level of the mixed Hodge structures that generalize the notion of ${\bf R}$-split mixed Hodge structure. This invariant can be expressed in terms of the Hodge numbers $h^{p,q}$ and some Hodge-type integers $s^{p,q}$ that are given by the relative position of the Hodge and conjugate filtrations. When the Hodge numbers are constant for a family of mixed Hodge structures, these numbers vary semicountinously. This allows to define stratifications of the bases of variations of mixed Hodge structures.