Non-abelian Local Invariant Cycles

Let f be a degeneration of Kähler manifolds. The local invariant cycle theorem states that for a smooth fiber of the degeneration, 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 Kähler manifolds is a proper map f from a Kähler manifold X onto the unit disk ∆ such that f is of maximum rank for all s ∈ ∆ except at the point s = 0. Let ∆∗ = ∆− {0}. We call Xt = f(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. Fix t ∈ ∆∗ and a base point x ∈ Xt once and for all. There is a monodromy action (see, for example, [3]) π1(∆ ∗)× H(Xt,C)→ H(Xt,C). The local invariant cycle theorem states that a cohomology class in H(Xt,C), fixed by the monodromy action, is a restriction of a cohomology class in H(X,C) [2, 9, 10]. Fix a generator T ∈ π1(∆) ∼= Z. Then T determines the monodromy action and gives rise to the isomorphisms T ∗ : H(Xt,C)→ H(Xt,C), T∗ : π1(Xt, x)→ π1(Xt, y). These isomorphisms are actually induced from a Picard-Lefschetz diffeomorphism which we shall also denote by