Classical and irregular Hodge numbers

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.