On the invariance of irregular Hodge numbers under crepant birational equivalences

Abstract

The Batyrev--Kontsevich theorem asserts that birational Calabi--Yau varieties have the same Hodge numbers. In this article, we prove an analogue for Landau--Ginzburg models $(U,f)$, where $U$ is a smooth quasi-projective complex variety and $f$ is a regular function on $U$. Under a natural non-degeneracy assumption, we show that the classes of such models in the localized Grothendieck ring of complex algebraic varieties with exponentials are invariant under crepant birational equivalences. Consequently, the irregular Hodge numbers of the twisted de Rham cohomology $\mathrm{H}^k_{\mathrm{dR}}(U,f)$ are invariant as well.