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.
We define quasi-coherent parabolic sheaves with real weights on a fine saturated log analytic space, and explain how to interpret them as quasi-coherent sheaves of modules on its Kato-Nakayama space. This recovers the description as sheaves on root stacks of arXiv:1001.0466 and arXiv:1410.1164 for rational weights, but also includes the case of arbitrary real weights.
We generalize the construction of moduli spaces of parabolic sheaves given by Maruyama and Yokogawa in [MY92] to the case of a projective fine saturated log scheme with a fixed global chart. Furthermore we construct moduli spaces of parabolic sheaves without fixing the weights.
We consider for structure groups ${\rm SU}(n)\,\subset\, {\rm SL}(n,\mathbb C)$ a densely defined toric structure on the moduli of framed parabolic sheaves on a three-punctured sphere, which degenerates to an actual toric structure. In combination with previous degeneration results, these extend to similar moduli for arbitrary Riemann surfaces.
The aim of these notes is to generalize Laumon's construction [18] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article "On the geometric Langlands conjecture" by Frenkel, Gaitsgory and Vilonen [10] carry over to our situation. We show that our sheaves descend to the moduli space of parabolic …
We construct geometric realizations of the r-colored bosonic and fermionic Fock space on the equivariant cohomology of the moduli space of framed rank r torsion-free sheaves on CP^2. Using these constructions, we realize geometrically all level one irreducible representations of the affine Lie algebra \hat{gl(r)}. The cyclic group Z_k acts naturally on the moduli space of sheaves, and the fixed-point components of this action are cyclic Nakajima quiver varieties . We realize level k irreducible representations of \hat{gl(r)} on the equivariant cohomology of these quiver varieties.
We show how the natural context for the definition of parabolic sheaves on a scheme is that of logarithmic geometry. The key point is a reformulation of the concept of logarithmic structure in the language of symmetric monoidal categories, which might be of independent interest. Our main result states that parabolic sheaves can be interpreted as quasi-coherent sheaves on certain stacks of roots.
This survey is based on my lectures given in last a few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorem, we show, without using vanishing theorem, a new observation that ${\rm dim}\,H^0(\sU_C,Θ_{\sU_C})$ is independent of all of the choices for any smooth curves. The estimate of various codimension and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.
We construct explicit elements $W_{ij}^k$ in (a completion of) the shifted quantum toroidal algebra of type $A$, and show that these elements act by 0 on the $K$-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements $W_{ij}^k$ will be related to $q$-deformed $W$-algebras of type $A$ for arbitrary nilpotent, which would imply a $q$-deformed version of the AGT correspondence between gauge theory with surface operators and conformal field theory.
Let $X$ be a complex torus of dimension $g$ and $\hat{X}$ be the dual torus. For any $g(g-1)/2$-tuple $λ$ of complex numbers of absolute value $1$, we define a non-commutative complex torus $X_λ$ as a sheaf of algebras on a real torus of dimension $g$. We prove that if all components of $λ$ are roots of unity, then the category of coherent sheaves on $X_λ$ is abelian and derived-equivalent to the category of coherent sheaves on $\hat{X}$ twisted by an element of the Brauer group of $\hat{X}$ determined by $λ$.
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 par…
We prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves (GPS) with a fixed determinant on a smooth projective curve are globally F-regular type. As an application, we prove vanishing theorems on the moduli spaces of semistable parabolic sheaves on a singular curve, which combining with Factorization theorems in [24] and [25] give two recurrence relations among dimensions of spaces of generalized theta functions. By using of these recurrence relations, we prove an explicit formula (Verlinde formula) for the dimension of spaces of generalized theta functions…
We calculate Betti numbers of the framed moduli space of instantons on $\hat{{\bf C}^2/{\bf Z}_2}$, under the assumption that the corresponding torsion free sheaves $E$ have vanishing properties ($Hom(E,E(-l_\infty))=Ext^2(E,E(-l_\infty))=0$). Moreover we derive the generating function of Betti numbers and obtain closed formulas. On the other hand, we derive a universal relation between the generating function of Betti numbers of the moduli spaces of stable sheaves on $X$ with an $A_1$-singularity and that on $\hat{X}$ blow-uped at the singularity, by using Weil conjecture. We call this the $O…
We generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langer's method to smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne-Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface $S$. In a joint a…
Let $X$ be a complex projective manifold. Fix two ample line bundles $H_0$ and $H_1$ on $X$. It is the aim of this note to study the variation of the moduli spaces of Gieseker semistable sheaves for polarizations lying in the cone spanned by $H_0$ and $H_1$. We attempt a new definition of walls which naturally describes the behaviour of Gieseker semistability. By means of an example, we establish the possibility of non-rational walls which is a substantially new phenomenon compared to the surface case. Using the approach of Ellingsrud and Goettsche via parabolic sheaves, we were able to show t…
We prove moduli spaces of semistable parabolic bundles and generalized parabolic sheaves with fixed determinant on a smooth projective curve are globally $F$-regular type.
We prove an analogue of Kac's Theorem, describing the dimension vectors of indecomposable coherent sheaves, or parabolic bundles, over weighted projective lines. We use a theorem of Peng and Xiao to associate a Lie algebra to the category of coherent sheaves for a weighted projective line over a finite field, and find elements of this Lie algebra which satisfy the relations defining the loop algebra of a Kac-Moody Lie algebra.
In this paper, we will look at actions on complex flag varieties $G/P$ of the torus $\hat{T}=\bigcap\limits_{α\inΔ\setminusΔ_P}\ker(α)$, and under reasonable assumptions, we will give a description of the set $X^{us}$ of unstable points for $\hat{T}$-linearized invertible sheaves. We will investigate the case where $P$ is a maximal parabolic subgroup, and show that $X^{us}$ can be written as a disjoint union of a Schubert variety and an opposite Schubert variety, and we deduce the vanishing of cohomology groups $H^i(Y,{\cal M})$ for invertible sheaves ${\cal M}$ on the quotient variety $Y$ for…
We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3 surfaces $\hat X$ can be chosen to be isomorphic to $X$; then the Fourier-Mukai transform is invertible, and the image of a zero-degree stable bundle $F$ is stable and has the same Euler characteristic as $F$.
We give a complete description of the two-dimensional moduli spaces of stable Higgs bundles of rank 2 over complex projective line with one irregular singular point, having a regular leading-order term, and endowed with a generic compatible parabolic structure such that the parabolic degree of the Higgs bundle is 0. Our method relies on elliptic fibrations of the rational elliptic surface, an equivalence of categories between irregular Higgs bundles and some sheaves on a ruled surface, and an analysis of stability conditions.
We show that there is an affine Schubert variety in the infinite dimensional partial Flag variety (associated to the two- step parabolic subgroup of the Kac-Moody group {\hat SL(n)}, corresponding to omitting α_0,α_d) which is a natural compactification of the cotangent bundle to the Grassmann variety.
We study the structure of a modified Fukaya category ${\frak F}(X)$ associated with a K3 surface $X$, and prove that whenever $X$ is an elliptic K3 surface with a section, the derived category of $\fF(X)$ is equivalent to a subcategory of the derived category ${\bold D}(\hat X)$ of coherent sheaves on the mirror K3 surface $\hat X$.
We define a new modular functor based on Kac-Wakimoto admissible representations and the corresponding D-module on the moduli space of rank 2 vector bundles with parabolic structure. A new fusion functor arises which is related to representation theory of the pair "osp(1|2),sl(2)" in the same way as the fusion functor for the Virasoro algebra is related to representation theory of the pair "sl(2),sl(2)"
Let G be an algebraic reductive group over a an algebraically closed field of positive characteristic. Choose a parabolic subgroup $P$ in $G$ and denote by $U$ its unipotent radical. Let $X$ be a $G$-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of l-adic sheaves on $X$ with respect to a generic character of $U$ commutes with Verdier duality. In the first example we take $χ$ to be an arbitrary $G$-variety and we prove the above property for all $\overline{P}$-equivariant sheaves on $X$ where $\overline{P}$ is an opposite parabolic s…
We globalize the derived version of the McKay correspondence of Bridgeland-King-Reid, proven by Kawamata in the case of abelian quotient singularities, to certain log algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible log blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under log blow-up, up to Morita equivalence.
The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of CYBE, based on torsion free coherent sheaves,…
