Wobbly moduli of chains, equivariant multiplicities and U(n₀,n₁)-Higgs bundles

We give a birational description of the reduced schemes underlying the irreducible components of the nilpotent cone and the C*-fixed point locus of length two in the moduli space of Higgs bundles. By producing criteria for wobbliness, we are able to determine wobbly fixed point components of type (n₀,n₁) and prove that these are precisely U(n₀,n₁)-wobbly components. We compute the virtual equivariant multiplicities of fixed points as defined by Hausel-Hitchin and find that they are polynomial for all partitions other than (2,1) and (4,3). In particular, this proves that they provide an obstruction to the existence of very stable fixed points only for very specific components.