Dehn Twists and Invariant Cocycles

A degeneration of compact Kähler manifolds gives rise to a monodromy action on Betti moduli space H (X,G) = Hom(π1(X), G)/G over smooth fibres with a complex algebraic structure group G being either abelian or reductive. Assume that the singularities of the central fibre is of normal crossing. When G = C, the invariant cocycles arise from the global cocycles. This is no longer true in general. In this paper, we produce large families of locally invariant cocycles that do not arise from global ones for reductive G. These examples exist even when G is abelian, as long as G contains multiple torsion points. Finally, for general G, we conjecture a new version of local invariant cocycle theorem and produce some suggestive examples.