On nefness of the lowest piece of Hodge modules

2025/07/02 by Ze Yun, Yun, Ze
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2507.01731

openalex publication_date 2025/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

Citations

• A Hodge-theoretic proof of Hwang's theorem on base manifolds of Lagrangian fibrations
• Unitary representations of real groups and localization theory for Hodge modules
• On semipositivity theorems
• Hodge modules on complex tori and generic vanishing for compact Kähler manifolds
• An overview of Morihiko Saito's theory of mixed Hodge modules
• Variations of mixed Hodge structure and semi-positivity theorems
• Characterization of Abelian Varieties

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• Flat Vector Bundles on Very General Curves and Codimension of Non-Abelian Hodge Loci
• Intersection theory of nef b-divisor classes
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• Generically nef vector bundles on ruled surfaces
• Nef cones of some Quot schemes on a smooth projective curve