Wobbly Fixed Points, Equivariant Multiplicities and U(p,q)-Higgs Bundles

Explore the geometry of Higgs bundle moduli spaces through an in-depth lecture on wobbly fixed points, equivariant multiplicities, and U(p,q)-Higgs bundles. Delve into the global nilpotent cone and its significance in governing the moduli space geometry. Examine the classification of fixed points into wobbly and very stable categories based on C* dynamics. Investigate Hausel and Hitchin's findings on the relationship between very stable fixed points and multiplicities of irreducible components of the nilpotent cone. Analyze nilpotent order two fixed points as a specific case of U(p,q)-Higgs bundles, and discover why they typically lack very stable points. Compare these results with the computation of virtual equivariant multiplicities. Conclude by exploring the equivalence between general wobbly points and U(p,q)-wobbly points, providing insights into this complex mathematical topic.