Wobbly moduli of chains, equivariant multiplicities and $\mathrm{U}(n_0,n_1)$-Higgs bundles

Abstract: We give a birational description of the reduced schemes underlying the irreducible components of the nilpotent cone and the $\mathbb{C}^\times$-fixed point locus of length two in the moduli space of Higgs bundles. We show that fixed point components of type $(n_0,n_1)$ are wobbly, and that they are also $\mathrm{U}(n_0,n_1)$-wobbly (an `a priori stronger notion). We analyse their virtual equivariant multiplicities and Euler pairings with downward flows for type $(1,\dots, 1)$ fixed points, to find that both invariants fail to fully detect all wobbly components for partitions other than $(2,1)$.