The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties

Abstract

Given any connected topological space X, assume that there exists an epimorphism $\phi {:}\; \pi _1(X) \rightarrow {\mathbb {Z}}$. The deck transformation group ${\mathbb {Z}}$ acts on the associated infinite cyclic cover $X^\phi$ of X, hence on the homology group $H_i(X^\phi , {\mathbb {C}})$. This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring ${\mathbb {C}}[t,t^{-1}]$, which is a finite dimensional ${\mathbb {C}}$-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When X is a compact Kähler manifold, we show that all the Jordan blocks are of size one. When X is a smooth complex quasi-projective variety, we give an upper bound on the sizes of the Jordan blocks, which is an analogue of the Monodromy Theorem for the local Milnor fibration.