Classical and irregular Hodge numbers

TL;DR

This paper establishes a clear relationship between irregular Hodge numbers of certain complex varieties and classical Hodge numbers, providing explicit formulas and demonstrating their invariance under specific conditions.

Contribution

It offers an explicit characterization of irregular Hodge numbers in terms of classical Hodge numbers, linking them to Hodge-theoretic numbers from Landau-Ginzburg models.

Findings

Irregular Hodge numbers are explicitly characterized via classical Hodge numbers.

Irregular Hodge numbers of non-degenerate functions are invariant under different choices.

A concrete formula for irregular Hodge numbers of unipotent non-degenerate functions is provided.

Abstract

Let $U$ be a smooth quasi-projective complex variety with a regular function $f$. The twisted de Rham cohomology groups $\mathrm{H}^k_{\mathrm{dR}}(U, f)$ carry the decreasing irregular Hodge filtration, whose graded pieces have dimensions known as the irregular Hodge numbers. In this paper, we prove that the irregular Hodge numbers admit an explicit characterization in terms of classical Hodge numbers, closely related to Hodge-theoretic numbers constructed by Katzarkov, Kontsevich, and Pantev for Landau--Ginzburg models. As direct applications, we show that irregular Hodge numbers of non-degenerate functions are independent of the choice of non-degenerate functions, and we give a concrete formula for irregular Hodge numbers for unipotent non-degenerate functions.