Hyperplane Arrangements and Mixed Hodge Numbers of the Milnor Fiber. (4th February 2021)

• Record Type: Journal Article
• Authors: Kutler, Max; Usatine, Jeremy
• Is Part Of: International mathematics research notices. Volume 2022:Number 11(2022)
• Page Start: 8644
• Page End: 8684
• Publication Date: 2021-02-04
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa393
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: For a complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$ . We prove that the map taking an arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of any simple normal crossing compactification of $F_{\mathcal{A}}$ . As a direct consequence, we obtain a combinatorial formula for the top weight cohomology of $F_{\mathcal{A}}$, recovering a result of Dimca and Lehrer.