Wild Hodge theory and Hitchin-Kobayashi correspondence [after T. Mochizuki]

TL;DR

This paper extends the Hitchin-Kobayashi correspondence to wild Hodge structures with irregular singularities, proving a conjecture related to the Hard Lefschetz theorem for holonomic D-modules.

Contribution

It develops a theory of wild Hodge structures with irregular singularities and generalizes the correspondence between flat bundles and Higgs bundles to this setting.

Findings

Constructs a theory of wild Hodge structures with irregular singularities.

Proves a conjecture of Kashiwara on the Hard Lefschetz theorem for holonomic D-modules.

Extends the correspondence of Corlette and Simpson to irregular cases.

Abstract

T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irregular singularities. As an application, he proves a conjecture of Kashiwara concerning a generalization of the Hard Lefschetz theorem when the coefficients are the de Rham complex of a simple holonomic D-module on smooth complex projective variety.