2021/05/12 by Saito, Morihiko
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2105.05770
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≥ 2 if X∖(X[(m)]∪ X⟨ 3⟩) or more generally X∖(X[(m)]∪ X⟨ 3⟩∪ Xd) is connected. Here X[(m)] is the set of points of X with multiplicity divisible by m, and X⟨ 3⟩:=\bigcupi,j,kXi∩ Xj∩ Xk with Xi the irreducible components of X, where the union is taken over i,j,k with \rm codim Xi∩ Xj∩ Xk=3. This hypothesis can be relaxed to some extent. The assertion is reduced to the case of a line arrangement in \bf P2 by Artin's vanishing theorem (where X⟨ 3⟩=∅), and we use a projection from \bf P2 to \bf P1 with center a sufficiently general point of Xd. It may be expected that the assumption of an improved assertion is always satisfied for m≥ 5 (and also for m=4 except the Hessian arrangement). The resulting vanishing of eigenspaces has been conjectured for m≥ 5.