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A V_{12} Fano threefold is a smooth Fano threefold X of index 1 with Pic X = Z and (-K_X)^3=12. We show that the bounded derived category of coherent sheaves on any V_{12} threefold X admits a semiorthogonal decomposition consisting of two exceptional bundles and of the derived category of a curve of genus 7. As an application we show that the Fano surface of X (the surface parameterizing conics on X) is canonically isomorphic to the symmetric square of the associated genus 7 curve., Comment: 11 pages

We show that the projectivization of the exceptional rank 2 vector bundle on an arbitrary smooth V14 Fano threefold after a certain natural flop turns into the projectivization of an instanton vector bundle on a smooth cubic threefold. And vice versa, starting from a smooth cubic threefold with an instanton vector bundle of charge 2 on it we reconstruct V14 threefold. Relying on the geometric properties of the above correspondence we prove that the orthogonals to the exceptional pairs in the bounded derived categories of coherent sheaves on a smooth V14 threefold and on the corresponding cubic threefold are equivalent as triangulated categories., Comment: 24 pages, dedicated to the memory of Andrei Nikolaevich Tyurin

Let the group G of m-th roots of unity act on the complex line by multiplication, inducing an action on the algebra, Diff, of polynomial differential operators on the line. Following Crawley-Boevey and Holland, we introduce a multiparameter deformation, D_c, of the smash-product (Diff # G). Our main result provides natural bijections between (roughly speaking) the following spaces: (1) G-equivariant version of Wilson's adelic Grassmannian of rank r; (2) Rank r projective D_c-modules (equipped with generic trivialization); (3) Rank r torsion-free sheaves on a `noncommutative quadric'; (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 math.AG/0103068. The resulting bijection between (1) and (4) reduces, in the very special case: r=1 and G=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 Wilson's result., Comment: 35pp, several remarks, and an open problem (at the end of the Introduction) added

We introduce an Uhlenbeck closure of the space of based maps from projective line to the Kashiwara flag scheme of an untwisted affine Lie algebra. For the algebra $\hat{sl}_n$ this space of based maps is isomorphic to the moduli space of locally free parabolic sheaves on $P^1\times P^1$ trivialized at infinity. The Uhlenbeck closure admits a resolution of singularities: the moduli space of torsion free parabolic sheaves on $P^1\times P^1$ trivialized at infinity. We compute the Intersection Cohomology sheaf of the Uhlenbeck space using this resolution of singularities. The moduli spaces of parabolic sheaves of various degrees are connected by certain Hecke correspondences. We prove that these correspondences define an action of $\hat{sl}_n$ in the cohomology of the above moduli spaces., Comment: 35pp., LaTeX

In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, we show that the $\Gamma$-equivariant Hilbert scheme $X^{\Gamma[n]}$ and the Hilbert scheme $X_\Gamma^{[n]}$ (where $X=\C^2$, $\Gamma\subset SL(\C^2)$ is a finite subgroup, and $X_\Gamma$ is a minimal resolution of $X/\Gamma$) are quiver varieties for the affine Dynkin graph, corresponding to $\Gamma$ via the McKay correspondence, the same dimension vectors, but different parameters $\zeta$ (for earlier results in this direction see [4, 12, 13]). In particular, it follows that the varieties $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$ are diffeomorphic. Computing their cohomology (in the case $\Gamma=\Z/d\Z$) via the fixed points of $(\C^*\times\C^*)$-action we deduce the following combinatorial identity: the number $UCY(n,d)$ of uniformly coloured in d colours Young diagrams consisting of nd boxes coincides with the number $CY(n,d)$ of collections of d Young diagrams with the total number of boxes equal to n., Comment: LaTeX, 27 pages

To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category Coh_t should be thought of as the category of coherent sheaves on a `noncommutative projective 2-space', equipped with a framing at the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coh_t and the Nakajima quiver variety arising from G via the McKay correspodence. We apply the above isomorphism to deduce generalized Crawley-Boevey & Holland conjecture, saying that the moduli space of `rank 1' projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for G=1, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, A_1, by points of the Calogero-Moser space, due to Cannings-Holland and Berest-Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on the noncommutative projective 2-space. It is totally different from the one used by Berest-Wilson, involving $\tau$-functions., Comment: 36pp, Crawley-Boevey & Holland conjecture generalized and proved

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)., Comment: 56 pages, AMS-Latex. Dedicated to A.N.Tyurin on his 60th birthday. v2: A section about instantons on a q-deformed R^4 and the corresponding q-deformation of the ADHM construction has been added

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 $K_\alpha$ of rank $n$ parabolic sheaves satisfying certain conditions. The top dimensional irreducible components of $K_\alpha$ are numbered by the isomorphism classes of $\alpha$-dimensional nilpotent representations of the cyclic quiver $\tilde{A}_{n-1}$. Summing up over all $\alpha\in{\Bbb N}[{\Bbb Z}/n{\Bbb Z}]$ we obtain a vector space $M$ with a basis of fundamental classes of top dimensional components of $K_\alpha$. 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., Comment: A reference added

Let $G$ be a semisimple complex Lie group with a Borel subgroup $B$. Let $X=G/B$ be the flag manifold of $G$. Let $C=P^1\ni\infty$ be the projective line. Let $\alpha\in H_2(X,{\Bbb Z})$. The moduli space of $G$-monopoles of topological charge $\alpha$ (see e.g. [Jarvis]) is naturally identified with the space $M_b(X,\alpha)$ of based maps from $(C,\infty)$ to $(X,B)$ of degree $\alpha$. The moduli space of $G$-monopoles carries a natural hyperk\"ahler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on $M_b(X,\alpha)$. It generalizes the well known formula for $G=SL_2$ (see e.g. [Atiyah-Hitchin]). Let $P\supset B$ be a parabolic subgroup. The construction of the Poisson structure on $M_b(X,\alpha)$ generalizes verbatim to the space of based maps $M=M_b(G/P,\beta)$. In most cases the corresponding map $T^*M\to TM$ is not an isomorphism, i.e. $M$ splits into nontrivial symplectic leaves. These leaves are explicilty described., Comment: v2: List of authors updated; v3: The formula for the symplectic form corrected; v4: Notations changed; v5: A few more corrections: final version

For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute its generating function. We define an action of the Langlands dual Lie algebra $g^L$ on this cohomology. We present a new geometric construction of the universal enveloping algebra $U(n^L_+)$ of the nilpotent subalgebra of $g^L$. It is realized in the Ext-groups of certain perverse sheaves on Quasimaps' spaces, and it is equipped with a canonical basis numbered by the irreducible components of certain algebraic cycles, isomorphic to the intersections of semiinfinite orbits in the affine Grassmannian of $G$. We compute the stalks of the IC-sheaves on the Schubert strata closures in $Q$. They carry a natural pure Tate Hodge structure, and their generating functions are given by the generic affine Kazhdan-Lusztig polynomials. In the Appendix we prove that Kontsevich's space of stable maps provides a natural resolution of $Q$., Comment: version published in AMS Translations

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 $\alpha\in N[I]$ of positive integers one can consider the space $Q_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_\alpha$ (Quasimaps), $Q^L_\alpha$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $\pi: Q^L_\alpha\to Q^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_\alpha$ on the Quasimaps' space $Q^D_\alpha$. Namely, we prove that this singular support $SS(IC_\alpha)$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $\pi: Q^L_\alpha\to Q^D_\alpha$., Comment: 8 pages, AmsLatex 1.1

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 $\alpha\in\BN[I]$ of positive integers one can consider the space $\CQ_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $\CB$. This space admits some remarkable compactifications $\CQ^D_\alpha$ (Quasimaps), $\CQ^L_\alpha$ (Quasiflags), $\CQ^K_\alpha$ (Stable Maps) of $\CQ_\alpha$ constructed by Drinfeld, Laumon and Kontsevich respectively. It has been proved that the natural map $\pi: \CQ^L_\alpha\to \CQ^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the cohomology $H^\bullet(\CQ^L_\alpha,\BQ)$ of Laumon's spaces or, equivalently, the Intersection Cohomology $H^\bullet(\CQ^L_\alpha,IC)$ of Drinfeld's Quasimaps' spaces. We calculate the generating function $P_G(t)$ (``Poincar\'e polynomial'') of the direct sum $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^D_\alpha,IC)$ 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., Comment: 21 pages, AmsLatex 1.1

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 $\alpha$ of positive integers one can consider the space $\MM\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $\FF$. This space has drawn much attention recently in connection with Quantum Cohomology. The space $\MM\alpha$ is smooth but not compact. The problem of compactification of $\MM\alpha$ proved very important. One compactification $\MML\alpha$ (the space of {\em quasiflags}), was constructed in \cite{L}. However, historically the first and most economical compactification $\MMD\alpha$ (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 $\pi:\MML\alpha\to\MMD\alpha$ 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 $\MMD\alpha$ 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})., Comment: Correction of misprints, alternative definition of Drinfeld's compactification included AMSLaTeX v 1.1