Topological computation of the first Milnor fiber cohomology of hyperplane arrangements

We study a topological method to calculate the first Milnor fiber cohomology of a defining polynomial of a reduced projective hyperplane arrangement $X$ of degree $d$. We show the vanishing of a monodromy eigenspace of the first Milnor fiber cohomology with eigenvalue of order $m\ge 2$ if $X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle})$ or more generally $X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle}\cup X_d)$ has at most two connected components, where we need some additional condition in the non-connected case. Here $X^{\langle 3\rangle}:=\bigcup_{i

Keywords:

• Cohomology
• center
• order
• Projection (relational algebra)
• Mathematics
• Topology
• Hyperplane
• Degree (graph theory)
• Monodromy
• Polynomial (hyperelastic model)

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