Dehn twists and invariant classes

TL;DR

This paper investigates invariant cohomology classes in Betti moduli spaces arising from degenerations of compact Kaehler manifolds, revealing new locally invariant classes not derived from global classes, especially for reductive groups.

Contribution

It introduces large families of locally invariant classes that are not global, expanding understanding of monodromy actions in degenerations of Kaehler manifolds.

Findings

Existence of locally invariant classes not from global classes for reductive G

Examples with G abelian containing multiple torsion points

A new conjecture on local invariant classes for general G

Abstract

A degeneration of compact Kaehler manifolds gives rise to a monodromy action on Betti moduli space H^1(X, G) = Hom(\pi_1(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 cohomology classes arise from the global classes. This is no longer true in general. In this paper, we produce large families of locally invariant classes 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 make a new conjecture on local invariant classes and produce some suggestive examples.