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

Cited by in corpus (16)

• Asymptotic behaviour of variation of pure polarized TERP structures
• Semi-positivity from Higgs bundles
• Foliations on the moduli space of rank two connections on the projective line minus four points
• Doubly periodic monopoles and $q$-difference modules
• Direct Images in Non Abelian Hodge Theory
• Perverse Leray filtration and specialization with applications to the Hitchin morphism
• Nilpotent orbits of a generalization of Hodge structures
• Rescalability of integrable mixed twistor $D$-modules
• Limits of families of Brieskorn lattices and compactified classifying spaces
• The Painlevé III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles
• $L^2$-representation of Hodge Modules
• The relative hermitian duality functor
• Poisson metrics and Higgs bundles over noncompact Kähler manifolds
• The Chern character of a parabolic bundle, and a parabolic Reznikov theorem in the case of finite order at infinity
• $L^2$-Dolbeault resolution of the lowest Hodge piece of a Hodge module
• Perverse sheaves and the reductive Borel-Serre compactification