On a Simplicial Complex Associated to the Monodromy

Abstract

Abstract.Suppose we have a complex analytic family, Vr \t\ < 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 / denote the identity map. We shall associate to the singular fibre a simplicial complex T, which is at most n-dimensional. Then under certain conditions on the family V, (which are satisfied for the Milnor fibration of an isolated singularity or if the V, are compact Kahler), there is an integer N > 0 such that (TN -¡fHk(V,) = 0 if and only if Hk(T) = 0.1. Introduction. 1.1. In this article, we let V be a complex hypersurface having normal crossings, where the complex dimension of V is «. Then we shall construct a simplicial complex T corresponding to V, where the real dimension of T is at most «. In fact, to each integer N > 0, we shall associate to F a complex TN, where Tx = T. Then we shall see how these complexes have applications in studying the monodromy about V.

A note on the dual complex of a resolution of singularities

“…and sends the vertex i r of the face E j i 0 ... is...i k to the vertex i r of the simplex ∆ j ′ i 0 ... is...i k . The complex Γ(E) was considered by G. L. Gordon in the paper [3].…”

“…gives a more complicated example (see [3]). If we blow up the origin, the exceptional divisor E ′ | X ′ consists of 3 lines E i , i = 1, 2, 3; every 2 of them intersect at a single point.…”

On the cohomology of a simple normal crossings divisor

“…Just think of the prime components D i as the vertices, non-empty 2-fold intersections D i ∩ D j as the edges, non-empty 3-fold intersections D i ∩ D j ∩ D k as the 2-faces, and so on. This simplicial complex, first studied by G. L. Gordon [9], is absent of any algebrao-geometric structure yet carries with it important skeletal information.…”

“…Inspired by the ideas of [17], we generalize and further develop the theory from the point of view of Stepanov, but note that similar avenues of thought have been explored by G. L. Gordon in the context of monodromy of complex analytic families [9], by P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan in the context of the cohomology of Kähler manifolds [5], by R. Friedman in the context of deformations and smoothings of varieties with normal crossings [7], and by P. Deligne [4], F. El Zein [6], J. Carlson [3], and J. H. M. Steenbrink [15] in the context of mixed Hodge structures. In Section 2, we briefly recall the notion of a presheaf on a simplicial complex and the cohomology of a simplicial complex with coefficients in a presheaf.…”

A note on resolution of rational and hypersurface singularities

“…The complex Γ(Z) was first studied by G. L. Gordon in connection to the monodromy in families (see [7]). We say that Γ(Z) is the dual complex associated to the resolution f .…”