On a simplicial complex associated to the monodromy

Abstract:

Suppose we have a complex analytic family, ${V_t}$, $\left | t \right | \leqslant 1$, such that the generic fibre is a nonsingular complex manifold of complex dimension n. Let T denote the monodromy induced from going once around the singular fibre and let I denote the identity map. We shall associate to the singular fibre a simplicial complex $\Gamma$, which is at most n-dimensional. Then under certain conditions on the family ${V_t}$ (which are satisfied for the Milnor fibration of an isolated singularity or if the ${V_t}$ are compact Kähler), there is an integer $N > 0$ such that ${({T^N} - I)^k}{H_k}({V_t}) = 0$ if and only if ${H_k}(\Gamma ) = 0$.