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 can show the vanishing of a monodromy eigenspace of the first Milnor fiber cohomology with eigenvalue of order $m\ge 2$ if $X\setminus (X^{[(m)]}\cup X^{⟨3⟩})$ or more generally $X\setminus (X^{[(m)]}\cup X^{⟨3⟩}\cup X_d)$ is connected. Here $X^{[(m)]}$ is the set of points of $X$ with multiplicity divisible by $m$, and $X^{⟨3⟩}:={\bigcup }_{i,j,k}{X}_i\cap X_j\cap X_k$ with $X_i$ the irreducible components of $X$, where the union is taken over $i,j,k$ with ${\rm codim},X_i\cap X_j\cap X_k=3$. This hypothesis can be relaxed to some extent. The assertion is reduced to the case of a line arrangement in ${\mathbf{P}}^2$ by Artin’s vanishing theorem (where $X^{⟨3⟩}=\mathrm{\varnothing }$), and we use a projection from ${\mathbf{P}}^2$ to ${\mathbf{P}}^1$ with center a sufficiently general point of $X_d$. It may be expected that the assumption of an improved assertion is always satisfied for $m\ge 5$ (and also for $m=4$ except the Hessian arrangement). The resulting vanishing of eigenspaces has been conjectured for $m\ge 5$.