Curvature of classifying spaces for Brieskorn lattices

TL;DR

This paper investigates the geometric structure of classifying spaces for Brieskorn lattices, establishing a hermitian metric and curvature estimates relevant to tt*-geometry and singularity theory.

Contribution

It introduces a canonical hermitian structure on the classifying space and provides curvature estimates, extending results from Hodge theory to Brieskorn lattices.

Findings

Existence of a canonical hermitian structure on the classifying space.

Curvature estimates analogous to those in Hodge theory.

Application to Fourier-Laplace transforms of Brieskorn lattices.

Abstract

We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the holomorphic sectional curvature of this hermitian metric, which is the analogue of a similar result for classifying spaces of pure polarized Hodge structures.