L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem

TL;DR

This paper investigates the growth and convergence of Hodge numbers and plurigenera in towers of abelian covers of irregular varieties, linking these to $L^2$-Betti numbers and the weak generic Nakano vanishing theorem.

Contribution

It introduces bounds for normalized Hodge numbers, computes $L^2$-Betti numbers under the weak generic Nakano vanishing condition, and extends Kollár's results on higher plurigenera multiplicativity.

Findings

Bounds for normalized Hodge numbers are sometimes optimal.

Computed $L^2$-Betti numbers for varieties satisfying the weak generic Nakano vanishing.

Extended Kollár's result to a broader class of varieties.

Abstract

We study the rate of growth of normalized Hodge numbers along a tower of abelian covers of a smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute the $L^2$-Betti numbers of irregular varieties that satisfy the weak generic Nakano vanishing theorem e.g., varieties with semismall Albanese map). Finally, we study the convergence of normalized plurigenera along towers of abelian covers of any irregular variety. As an application, we extend a result of Koll\'ar concerning the multiplicativity of higher plurigenera of a smooth projective variety of general type, to a wider class of varieties. In the Appendix, we study irregular varieties for which the first Betti number diverges along a tower of abelian covers induced by the Albanese variety.