$L^2$ representation of Simpson-Mochizuki's prolongation of Higgs bundles and the Kawamata-Viehweg vanishing theorem for semistable parabolic Higgs bundles

TL;DR

This paper develops an $L^2$-theoretic framework for prolongations of nilpotent harmonic bundles, providing new insights into their structure and applications to vanishing theorems in algebraic geometry.

Contribution

It introduces an $L^2$ fine resolution for Simpson-Mochizuki's prolongation of nilpotent harmonic bundles, extending previous results to a logarithmic setting and applying it to vanishing theorems.

Findings

Established an $L^2$-theoretic interpretation of the prolongation

Provided a new proof of the Kawamata-Viehweg vanishing theorem

Extended $L^2$ techniques to logarithmic Higgs bundles

Abstract

In this paper, we provide an $L^2$ fine resolution of the prolongation of a nilpotent harmonic bundle in the sense of Simpson-Mochizuki (an analytic analogue of the Kashiwara-Malgrange filtrations). This is the logarithmic analogue of Cattani-Kaplan-Schmid's and Kashiwara-Kawai's results on the $L^2$ interpretation of the intersection complex. As an application, we give an $L^2$-theoretic proof to the Nadel-Kawamata-Viehweg vanishing theorem with coefficients in a nilpotent Higgs bundle.