Non-abelian local invariant cycles

Abstract

Let f be a degeneration of Kahler manifolds. The lo- cal invariant cycle theorem states that for a smooth fiber of the de- generation, any cohomology class, invariant under the monodromy action, rises from a global cohomology class. Instead of the classical cohomology, one may consider the non-abelian cohomology. This note demonstrates that the analogous non-abelian version of the local invariant cycle theorem does not hold if the first non-abelian cohomology is the moduli space (universal categorical quotient) of the representations of the fundamental group. A degeneration of Kahler manifolds is a proper map f from a Kahler manifold X onto the unit disksuch that f is of maximum rank for all s ∈ � except at the point s = 0. Let � � = � − {0}. We call Xt = f 1 (Xt) a smooth fiber or generic fiber when t ∈ � � and X0 = f 1 (0) the singular or degenerated fiber. We assume the singularity in X0 is of normal crossing.