Vladimir Baranovsky

The variety of pairs of commuting nilpotent matrices is irreducible

In this paper we prove the dimension and the irreduciblity of the variety parametrizing all pairs of commuting nilpotent matrices. Our proof uses the connection between this variety and the punctual Hilbert scheme of a smooth algebraic surface.

Wilson's Grassmannian and a noncommutative quadric

Let the group μ_m of m th roots of unity act on the complex line by multiplication. This gives a μ_m-action on Diff, the algebra of polynomial differential operators on the line. Following Crawley-Boevey and Holland (1998), we introduce a multiparameter deformation Dτ of the smash product Diff #μ_m. Our main result provides natural bijections between (roughly speaking) the following spaces: (1) μ_m-equivariant version of Wilsons adelic Grassmannian of rank r ; (2) rank r projective Dτ-modules (with generic trivialization data); (3) rank r torsion-free sheaves on a “noncommutative quadric” ℙ^1 × τℙ^1; (4) disjoint union of Nakajima quiver varieties for the cyclic quiver with m vertices. The bijection between (1) and (2) is provided by a version of Riemann-Hilbert correspondence between D-modules and sheaves. The bijections between (2), (3), and (4) were motivated by our previous work Quiver varieties and a noncommutative ℙ^2 (2002). The resulting bijection between (1) and (4) reduces, in the very special case: r=1 and μ_m={1}, to the partition of (rank 1) adelic Grassmannian into a union of Calogero-Moser spaces discovered by Wilson. This gives, in particular, a natural and purely algebraic approach to Wilsons result (1998).

Orbifold Cohomology as Periodic Cyclic Homology

It is known from the work of Feigin–Tsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite group G acting on X the same procedure applied to G-equivariant sheaves gives the orbifold cohomology of X/G. As an application, in some cases we are able to obtain simple proofs of an additive isomorphism between the orbifold cohomology of X/G and the usual cohomology of its crepant resolution (the equality of Euler and Hodge numbers was obtained earlier by various authors). We also state some conjectures on the product structures, as well as the singular case; and a connection with a recent work by Kawamata.

On equivalences of derived and singular categories

AbstractLet X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X →

Representations of Quantum Tori and G-bundles on Elliptic Curves

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Moduli of Sheaves on Surfaces and Action of the Oscillator Algebra

, g:Y →

Quiver varieties and a noncommutative P2

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