arXiv subjects

Alexander Kuznetsov

Publications and source records attributed to Alexander Kuznetsov.

95 records · Page 6Linked to original sources

arXiv2000

Noncommutative Instantons and Twistor Transform

Recently N.Nekrasov and A.Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of the 4-dimensional real affine space. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative projective plane, certain complexes of sheaves on a noncommutative 3-dimensional projective space, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative projective plane has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative projective plane. The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative R^4 than the one considered by Nekrasov and Schwarz (a q-deformed R^4).

hep-th↗

arXiv1999

Parabolic sheaves on surfaces and affine Lie algebra $\hat{gl}_n$

We give an example of geometric construction (via Hecke correspondences) of certain representations of the affine Lie algebra $\hat{gl}_n$. The construction is similar to the one of [FK] for the Lie algebra $sl_n$. Given a surface with a smooth embedded curve $C$ we consider the moduli spaces $K_α$ of rank $n$ parabolic sheaves satisfying certain conditions. The top dimensional irreducible components of $K_α$ are numbered by the isomorphism classes of $α$-dimensional nilpotent representations of the cyclic quiver $\tilde{A}_{n-1}$. Summing up over all $α\in{\Bbb N}[{\Bbb Z}/n{\Bbb Z}]$ we obtain a vector space $M$ with a basis of fundamental classes of top dimensional components of $K_α$. The natural correspondences give rise to the action of Chevalley generators $e_i,f_i\in\hat{sl}_n$ on $M$. We compute explicitly the matrix coefficients of $e_i,f_i$ in the above basis. The central charge of $M$ depends on the genus of the curve $C$ and the degree of its normal bundle.

math.AG↗

arXiv1997

The Singular Supports of IC sheaves on Quasimaps' Spaces are Irreducible

Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $α\in N[I]$ of positive integers one can consider the space $Q_α$ of algebraic maps of degree $α$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_α$ (Quasimaps), $Q^L_α$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $π: Q^L_α\to Q^D_α$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_α$ on the Quasimaps' space $Q^D_α$. Namely, we prove that this singular support $SS(IC_α)$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $π: Q^L_α\to Q^D_α$.

alg-geom↗

arXiv1997

Global Intersection Cohomology of Quasimaps' Spaces

Let $C$ be a smooth projective curve of genus 0. Let $\CB$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $α\in\BN[I]$ of positive integers one can consider the space $\CQ_α$ of algebraic maps of degree $α$ from $C$ to $\CB$. This space admits some remarkable compactifications $\CQ^D_α$ (Quasimaps), $\CQ^L_α$ (Quasiflags), $\CQ^K_α$ (Stable Maps) of $\CQ_α$ constructed by Drinfeld, Laumon and Kontsevich respectively. It has been proved that the natural map $π: \CQ^L_α\to \CQ^D_α$ is a small resolution of singularities. The aim of the present note is to study the cohomology $H^\bullet(\CQ^L_α,\BQ)$ of Laumon's spaces or, equivalently, the Intersection Cohomology $H^\bullet(\CQ^L_α,IC)$ of Drinfeld's Quasimaps' spaces. We calculate the generating function $P_G(t)$ (``Poincaré polynomial'') of the direct sum $\oplus_{α\in\BN[I]}H^\bullet(\CQ^D_α,IC)$ and construct a natural action of the Lie algebra ${\frak{sl}}_n$ on this direct sum by some middle-dimensional correspondences between Quasiflags' spaces. We conjecture that this module is isomorphic to distributions on nilpotent cone supported at nilpotent subalgebra.

alg-geom↗

arXiv1996

The Laumon's resolution of Drinfeld's compactification is small

Let $C$ be a smooth projective curve of genus 0. Let $\FF$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $α$ of positive integers one can consider the space $\MMα$ of algebraic maps of degree $α$ from $C$ to $\FF$. This space has drawn much attention recently in connection with Quantum Cohomology. The space $\MMα$ is smooth but not compact. The problem of compactification of $\MMα$ proved very important. One compactification $\MMLα$ (the space of {\em quasiflags}), was constructed in \cite{L}. However, historically the first and most economical compactification $\MMDα$ (the space of {\em quasimaps}) was constructed by Drinfeld (early 80-s, unpublished). The latter compactification is singular, while the former one is smooth. Drinfeld has conjectured that the natural map $π:\MMLα\to\MMDα$ is a small resolution of singularities. In the present note we prove this conjecture. As a byproduct, we compute the stalks of $IC$ sheaf on $\MMDα$ and, moreover, the Hodge structure in these stalks. Namely, the Hodge structure is a pure Tate one, and the generating function for the $IC$ stalks is just the Lusztig's $q$-analogue of Kostant's partition function (see \cite{Lu}).

alg-geom↗