On nefness of the lowest piece of Hodge modules

TL;DR

This paper establishes degree lower bounds for quotient line bundles of the lowest piece of Hodge modules, explaining nefness failure when monodromies are non-unipotent, and recovers Kawamata's semi-positivity theorem.

Contribution

It provides new degree bounds depending on local monodromies, extending semi-positivity results to non-unipotent cases, with algebraic proofs and geometric examples.

Findings

Lower bounds depend on local monodromies and intersection numbers.

Nefness fails when monodromies at infinity are not unipotent.

Achievability of bounds demonstrated through geometric examples.

Abstract

We give degree lower bounds for quotient line bundles of the lowest piece of a Hodge module induced by a complex variation of Hodge structures outside a simple normal crossing divisor, beyond the unipotent variation case. This note aims to explain the failure of nefness when the monodromies at infinity are not unipotent. The lower bounds depend on local monodromies at infinity and intersection numbers with the boundary divisors. In particular it recovers Kawamata's semi-positivity theorem for unipotent variations. The proof is algebraic via a vanishing theorem for twisted Hodge modules. We also give geometric examples to show that the lower bound can be achieved.