Vanishing of Local Cohomology with Applications to Hodge Theory

2022/11/10 by Scott Hiatt, Hiatt, Scott
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2211.05876

openalex publication_date 2022/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H = ((H, F^\bullet), L) be a polarized variation of Hodge structure on a smooth quasi-projective variety U. By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure H can be viewed as a polarized Hodge module M ∈ HM(U). Let X be a compactification of U, and j:U \hookrightarrow X is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study j₊M ∈ D^bMHM(X). In particular, we study the graded pieces of the de Rham complex Gr^F_pDR(j₊M) ∈ D^b_coh(X), and the Hodge structure of Hⁱ(U,L) for i in sufficiently low degrees.

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