Torsion-free Sheaves Research Articles

On Discrete Orthogonal Coordinates Corresponding to Torsion-Free Sheaves over Reducible Spectral Curves

On Discrete Orthogonal Coordinates Corresponding to Torsion-Free Sheaves over Reducible Spectral Curves

Irreducibility of moduli of vector bundles over a very general sextic surface

Let S be a very general smooth hypersurface of degree 6 in P3. In this paper we will prove that the moduli space of μ-stable rank 2 torsion free sheaves with respect to hyperplane section having c1=OS(1), with fixed, c2≥27 is irreducible.

Stability of kernel sheaves associated to rank one torsion-free sheaves

Stability of kernel sheaves associated to rank one torsion-free sheaves

Virasoro constraints for moduli of sheaves and vertex algebras

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce’s vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only (p,p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,p)$\\end{document} cohomology classes by reducing the statements to the rank 1 case.

Orthogonal Curvilinear Coordinate Systems and Torsion-Free Sheaves over Reducible Spectral Curves

Orthogonal Curvilinear Coordinate Systems and Torsion-Free Sheaves over Reducible Spectral Curves

Recurrence relation for instanton partition function in SU(N) gauge theory

We derive a residue formula and as a consequence a recurrence relation for the instanton partition function in mathcal{N} = 2 supersymmetric theory on ℂ2 with SU(N) gauge group.The particular cases of SU(2) and SU(3) gauge groups were considered in the literature before. The recurrence relation with SU(2) gauge group is long well known and was found as the Alday-Gaiotto-Tachikawa (AGT) counterpart of the Zamolodchikov relation for the Virasoro conformal blocks. In the SU(3) case a residue formula for the term with the minimal number of instantons was found and basing on it a recurrence relation was conjectured.We give a complete proof of the residue formula in all instanton orders in presence of any number of matter hypermultiplets in the adjoint and fundamental representations. The recurrence relation however describes only theories with not too much matter hypermultiplets so that the behaviour at infinity is moderate. The guideline of the proof is an algebro-geometric interpretation of the mathcal{N} = 2 supersymmetric gauge theory partition function in terms of the framed torsion-free sheaves. Lead by this interpretation we formulate a refined version of the residue formula and prove it by direct algebraic manipulations.

On the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces

AbstractThe aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;\,c_1,c_2)$ to $Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ .

Anticanonical geometry of the blow-up of mathbb {P}^4 in 8 points and its Fano model

Building on the work of Casagrande–Codogni–Fanelli, we develop our study on the birational geometry of the Fano fourfold Y=M_{S,-K_S} which is the moduli space of semi-stable rank-two torsion-free sheaves with c_1=-K_S and c_2=2 on a polarised degree-one del Pezzo surface (S,-K_S). Based on the relation between Y and the blow-up of mathbb {P}^4 in 8 points, we describe completely the base scheme of the anticanonical system |{-}K_Y|. We also prove that the Bertini involution iota _Y of Y, induced by the Bertini involution iota _S of S, preserves every member in |{-}K_Y|. In particular, we establish the relation between iota _Y and the anticanonical map of Y, and we describe the action of iota _Y by analogy with the action of iota _S on S.

Langton’s Type Theorem on Algebraic Orbifolds

In this paper, we prove the Langton’s type theorem on separatedness and properness for the moduli functor of torsion free semistable sheaves on algebraic orbifolds over an algebraically closed field k.

Classifying the irreducible components of moduli stacks of torsion free sheaves on K3 surfaces and an application to Brill-Noether theory

In this article, we classify the irreducible components of moduli stacks of torsion free sheaves of rank 2 on K3 surfaces of Picard number 1. For ruled surfaces, the components of moduli stacks of torsion free sheaves were classified by Walter ([18]). Moreover, by virtue of our result, we classify the irreducible components of Brill-Noether loci of Hilbert schemes of points on K3 surfaces.

Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincaré polynomial of the associated cohomology ring.

Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic threefolds

We focus on a class of Weierstraß elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically K-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform of a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We also show that the inverse Fourier-Mukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to modification in codimension 2.

Irreducible components of the moduli space of rank 2 sheaves of odd determinant on projective space

We describe new irreducible components of the moduli space of rank 2 semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes (c1,c2,c3)=(−1,2n,0) and (c1,c2,c3)=(0,n,0) always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, and compute the exact number of irreducible components of the moduli spaces of rank 2 semistable torsion free sheaves with Chern classes (c1,c2,c3)=(−1,2,m) for non negative values for m; all components turn out to be rational. Furthermore, we also prove that these moduli spaces are connected, showing that some of sheaves here considered are smoothable.

Urod algebras and Translation of W-algebras

Abstract In this work, we introduce Urod algebras associated to simply laced Lie algebras as well as the concept of translation of W-algebras. Both results are achieved by showing that the quantum Hamiltonian reduction commutes with tensoring with integrable representations; that is, for V and L an affine vertex algebra and an integrable affine vertex algebra associated with $\mathfrak {g}$ , we have the vertex algebra isomorphism $H_{DS,f}^{0}(V\otimes L)\cong H_{DS,f}^{0}(V)\otimes L$ , where in the left-hand-side the Drinfeld–Sokolov reduction is taken with respect to the diagonal action of $\widehat {\mathfrak {g}}$ on $V{\otimes } L$ . The proof is based on some new construction of automorphisms of vertex algebras, which may be of independent interest. As corollaries, we get fusion categories of modules of many exceptional W-algebras, and we can construct various corner vertex algebras. A major motivation for this work is that Urod algebras of type A provide a representation theoretic interpretation of the celebrated Nakajima–Yoshioka blowup equations for the moduli space of framed torsion free sheaves on $\mathbb {CP}^{2}$ of an arbitrary rank.

Complex algebraic compactifications of the moduli space of Hermitian Yang–Mills connections on a projective manifold

In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. In the present article, following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge theoretic compactification by adding certain ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves allows us to endow the gauge theoretic compactification with the structure of a complex analytic space.

Framed sheaves on projective space and Quot schemes

We prove that, given integers mge 3, rge 1 and nge 0, the moduli space of torsion free sheaves on {mathbb {P}}^m with Chern character (r,0,ldots ,0,-n) that are trivial along a hyperplane D subset {mathbb {P}}^m is isomorphic to the Quot scheme mathrm{Quot}_{{mathbb {A}}^m}({mathscr {O}}^{oplus r},n) of 0-dimensional length n quotients of the free sheaf {mathscr {O}}^{oplus r} on {mathbb {A}}^m. The proof goes by comparing the two tangent-obstruction theories on these moduli spaces.

Generalization of a conjecture of Mumford

A conjecture of Mumford predicts a complete set of relations between the generators of the cohomology ring of the moduli space of rank 2 semi-stable sheaves with fixed odd degree determinant on a smooth, projective curve of genus at least 2. The conjecture was proven by Kirwan in 1992 [24]. In this article, we generalize the conjecture to the case when the underlying curve is irreducible, nodal. In fact, we show that these relations (in the nodal curve case) arise naturally as degeneration of the Mumford relations shown by Kirwan in the smooth curve case. As a byproduct, we compute the Hodge-Poincaré polynomial of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed determinant on an irreducible, nodal curve.

Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

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, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.

The spectrum of torsion free sheaves on P3 and applications

The spectrum of torsion free sheaves on P3 and applications

Toric Sheaves on Hirzebruch Orbifolds

We provide a stacky fan description of the total space of certain split vector bundles, as well as their projectivization, over toric Deligne-Mumford stacks. We then specialize to the case of Hirzebruch orbifold $\mathcal{H}_{r}^{ab}$ obtained by projectivizing $\mathcal{O} \oplus \mathcal{O}(r)$ over the weighted projective line $\mathbb{P}(a,b)$. Next, we give a combinatorial description of toric sheaves on $\mathcal{H}_{r}^{ab}$ and investigate their basic properties. With fixed choice of polarization and a generating sheaf, we describe the fixed point locus of the moduli scheme of $\mu$-stable torsion free sheaves of rank $1$ and $2$ on $\mathcal{H}_{r}^{ab}$. As an example, we obtain explicit formulas for generating functions of Euler characteristics of locally free sheaves of rank 2 on $\mathbb{P}(1,2) \times \mathbb{P}^1$.