Operator ideals in Tate objects

Tate's central extension originates from 1968 and has since found many applications to curves. In the 80s Beilinson found an n-dimensional generalization: cubically decomposed algebras, based on ideals of bounded and discrete operators in ind-pro limits of vector spaces. Kato and Beilinson independently defined '(n-)Tate categories' whose objects are formal iterated ind-pro limits in general exact categories. We show that the endomorphism algebras of such objects often carry a cubically decomposed structure, and thus a (higher) Tate central extension. Even better, under very strong assumptions on the base category, the n-Tate category turns out to be just a category of projective modules over this type of algebra

Hilbert Schemes as Moduli of Higgs Bundles and Local Systems

We construct five families of 2D moduli spaces of parabolic Higgs bundles (respectively, local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve (respectively, twisted cotangent bundle). We show that the Hilbert scheme of m points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively, local systems), confirming a conjecture of Boalch in these cases and extending a result of Gorsky-Nekrasov-Rubtsov. Using the McKay correspondence, we establish the autoduality conjecture for the derived categories of the moduli spaces of Higgs bundles under consideratio