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

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.

We give the correspondence of semisimple D-modules and polarizable pure twistor D-modules in the sense of Sabbah. As a result, the regular holonomic version of Kashiwara's conjecture is established. The main difference of the versions are explained in the subsection 1.6

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