Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules

TL;DR

This paper investigates the asymptotic properties of tame harmonic bundles, establishing their structure and applications to $D$-modules, including norm estimates and a proof of Sabbah's conjecture.

Contribution

It proves local freeness of prolongations, constructs polarized mixed twistor structures, and establishes a correspondence between semisimple regular holonomic $D$-modules and pure twistor $D$-modules, confirming Sabbah's conjecture.

Findings

Proved local freeness of prolongations of tame harmonic bundles.

Established the polarized mixed twistor structure.

Confirmed the conjecture relating $D$-modules and twistor structures.

Abstract

We study the asymptotic behaviour of tame harmonic bundles. First of all, we prove a local freeness of the prolongation by an increasing order. Then we obtain the polarized mixed twistor structure. As one of the applications, we obtain the norm estimate of holomorphic or flat sections by weight filtrations of the monodromies. As other application, we establish the correspondence of semisimple regular holonomic $D$-modules and polarizable pure imaginary pure twistor $D$-modules through a tame pure imaginary harmonic bundles, which is a conjecture of Sabbah. Then the regular holonomic version of Kashiwara's conjecture follows from the results of Sabbah and us. Keywords: Higgs fields, harmonic bundle, variation of Hodge structure, mixed twistor structure, $D$-module.