Fourier-Laplace transform of a variation of polarized complex Hodge structure, II

TL;DR

This paper establishes a connection between the limit of a supersymmetric index and the spectral polynomial for Fourier-Laplace transforms of polarized Hodge structures, and extends Hodge filtration definitions with new degeneration results.

Contribution

It proves the equality of the limit of the supersymmetric index and the spectral polynomial, and extends Deligne's Hodge filtration with an E_1 degeneration property.

Findings

Limit of supersymmetric index equals spectral polynomial.

Extended Hodge filtration with E_1 degeneration.

Established new links between Fourier-Laplace transforms and Hodge theory.

Abstract

We show that the limit, by rescaling, of the `new supersymmetric index' attached to the Fourier-Laplace transform of a polarized variation of Hodge structure on a punctured affine line is equal to the spectral polynomial attached to the same object. We also extend the definition by Deligne of a Hodge filtration on the de Rham cohomology of a exponentially twisted polarized variation of complex Hodge structure and prove a E_1 degeneration property for it.