On Betti Numbers of Milnor Fiber of Hyperplane Arrangements

TL;DR

This paper establishes a combinatorial upper bound for the first characteristic polynomial of the Milnor fiber of central hyperplane arrangements, advancing understanding of its Betti numbers and monodromy properties.

Contribution

It introduces a new combinatorial upper bound for the Milnor fiber's first characteristic polynomial, improving previous results and providing obstructions for trivial monodromy.

Findings

Derived a combinatorial upper bound for the first characteristic polynomial.

Identified a combinatorial obstruction for trivial algebraic monodromy.

Validated results through calculations and comparisons with known examples.

Abstract

Let $\mathcal{A}$ be a central hyperplane arrangement in $\mathbb{C}^{n+1}$ and $H_i,i=1,2,...,d$ be the defining equations of the hyperplanes of $\mathcal{A}$. Let $f=\prod_i H_i$. There is a global Milnor fibration $F\hookrightarrow \mathbb{C}^{n+1} \setminus \mathcal{A} \xrightarrow{f} \mathbb{C}^*,$ where $F$ is called the Milnor fiber and can be identified as the affine hypersurface $f=1$ in $\mathbb{C}^{n+1}$. Many open questions have been raised subject to $F$. In particular, it has been conjectured that the integral homology, or the characteristic polynomial, hence the Betti numbers, of $F$ are also determined by the intersection lattice $L(\mathcal{A})$. In this paper, we find a combinatorial upper bound for the first the characteristic polynomial of the Milnor fiber for central hyperplane arrangements, which improves existing results in the study. As a corollary, we obtain a combinatorial obstruction for trivial algebraic monodromy of the first homology of Milnor fiber. Calculations and comparisons to known examples computed using Fox calculus will be provided.