On Betti Numbers of Milnor Fiber of Hyperplane Arrangements

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 $F\hookrightarrow {\mathbb{C}}^{n+1}\setminus \mathcal{A}\stackrel{f}{\to }{\mathbb{C}}^{\ast },$ 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.

10 pages, 2 figures