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

2003/12/11 by Takuro Mochizuki, Mochizuki, Takuro · 14 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG #math.DG #msc:14C30 #msc:32S40 #msc:53C43

paper · pdf · doi:10.48550/arxiv.math/0312230

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

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.

Cited by

• Hyperbolicity and fundamental groups of complex quasi-projective varieties (II): via non-abelian Hodge theories
• Asymptotic behaviour of variation of pure polarized TERP structures
• The Hodge theory of the Decomposition Theorem (after de Cataldo and Migliorini)
• Harmonic maps in singular geometry and rigidity
• Kobayashi-Hitchin correspondence for analytically stable bundles
• Absolute sets and the Decomposition Theorem
• A generalization of Barannikov-Kontsevich theorem
• Generic vanishing theory via mixed Hodge modules
• Semi-positivity from Higgs bundles
• Rescalability of integrable mixed twistor D-modules
• Vanishing theorem for tame harmonic bundles via L²-cohomology
• Application of Cheeger-Gromov theory to the l²-cohomology of harmonic Higgs bundles over covering of finite volume complete manifolds
• L²-representation of Hodge Modules
• Mixed Hodge modules and mixed twistor modules

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