Free Sheaves Research Articles

Stability of Gieseker stable sheaves on K3 surfaces in the sense of Bridgeland and some applications

We show that some Gieseker stable sheaves on a projective K3 surface $X$ are stable with respect to a stability condition of Bridgeland on the derived category of $X$ if the stability condition is in explicit subsets of the space of stability conditions depending on the sheaves. Furthermore we shall give two applications of the result. As a part of these applications, we show that the fine moduli space of Gieseker stable torsion free sheaves on a K3 surface with Picard number one is the moduli space of $\mu$-stable locally free sheaves if the rank of the sheaves is not a square number.

Nonabelian Jacobian of smooth projective surfaces-a survey

The nonabelian Jacobian J(X; L, d) of a smooth projective surface X is inspired by the classical theory of Jacobian of curves. It is built as a natural scheme interpolating between the Hilbert scheme X [d] of subschemes of length d of X and the stack M X (2, L, d) of torsion free sheaves of rank 2 on X having the determinant O X (L) and the second Chern class (= number) d. It relates to such influential ideas as variations of Hodge structures, period maps, nonabelian Hodge theory, Homological mirror symmetry, perverse sheaves, geometric Langlands program. These relations manifest themselves by the appearance of the following structures on J(X; L, d): 1) a sheaf of reductive Lie algebras 2) (singular) Fano toric varieties whose hyperplane sections are (singular) Calabi-Yau varieties 3) trivalent graphs. This is an expository paper giving an account of most of the main properties of J(X; L, d) uncovered in Reider 2006 and ArXiv:1103.4794v1.

Note on the Serre‐Swan theorem

AbstractWe determine a class of ringed spaces,\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(X,\mathcal {O}_{X})$\end{document}for which the category of locally free sheaves of bounded rank overXis equivalent to the category of finitely generated projective\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Gamma ({X},{\mathcal {O}_{X}})$\end{document}‐modules. The well‐known Serre‐Swan theorems for affine schemes, differentiable manifolds, Stein spaces, etc., are then derived.

Homogeneous components in the moduli space of sheaves and Virasoro characters

The moduli space M(r,n) of framed torsion free sheaves on the projective plane with rank r and second Chern class equal to n has the natural action of the (r+2)-dimensional torus. In this paper, we look at the fixed point set of different one-dimensional subtori in this torus. We prove that in the homogeneous case the generating series of the numbers of the irreducible components has a beautiful decomposition into an infinite product. In the case of odd r, these infinite products coincide with certain Virasoro characters. We also propose a conjecture in a general quasihomogeneous case.

Stacks of trigonal curves

In this paper we study the stack Tg of smooth triple covers of a conic; when g � 5 this stack is embedded Mg as the locus of trigonal curves. We show that Tg is a quotient (Ug/ g), where g is a certain algebraic group and Ug is an open subscheme of a g-equivariant vector bundle over an open subscheme of a representation of g. Using this, we compute the integral Picard group of Tg when g > 1. The main tools are a result of Miranda that describes a flat finite triple cover of a scheme S as given by a locally free sheaf E of rank two on S, with a section of Sym 3 E det E _ , and a new description of the stack of globally generated locally free sheaves of fixed rank and degree on a projective line as a quotient stack. In moduli theory, stacks have often been constructed as quotients by group actions. If an algebraic stack F is a quotient stack (X=G), where G is an algebraic group acting on an algebraic variety X, the geometry of F is the geometry of the action of G on X, and one can apply to F the powerful techniques that have been developed for studying invariants of group actions in algebraic topology and algebraic geometry. Even just knowing that such a presentation exists, even without an explicit description of the action, can be useful: but it is even better when the variety X and the group G are fairly simple, so that this description may be used directly to study F. This does not seem possible in many cases: for example, the stack Mg of smooth curves of genus g is of the form (X=G), but when g is large the space X is complicated, and to our knowledge no general result about Mg has been proved by exploiting this presentation. (Of course, in Teichmuller theory one represents Mg as a quotient of an action of the Teichmuller group, which is an infinite discrete group, acting on a ball in C 3g−3 , but this description is topological, and it is hard to use it directly to prove algebraic geometric results about Mg.) It has long been known that in characteristic different from 2 and 3, the stack M1;1 of elliptic curves is a quotient ((X=Gm)), where X is the complement of the hypersurface 4x 3 + 27y 2 = 0 in A 2 , and Gm acts with weights 4 and 6. This gives an easy proof of the fact, due to Mumford ((Mum65)), that the Picard group of M1;1 is cyclic of order 12. In (Vis98), the second author gives a presentation of M2 as a quotient (X=GL2), where X is the scheme of smooth binary forms of degree 6 in two variables (the action of GL2 on X is a twist of the customary one). As an application he computes the Chow ring of M2. This was generalized in (AV04) to the stack Hg of hyperelliptic curves of genus g, which has a presentation as a quotient (Xg=GL2) (if g is even), or (Xg=(GmPGL2)) (if g is odd), where X is the space of smooth binary forms of degree 2g + 2 in two variables; this allows

Absolutely split locally free sheaves on Brauer–Severi varieties of index two

Let X be a Brauer–Severi variety over a field k associated with a central simple k -algebra of index two. This variety has the property of being isomorphic to a projective space P L N after base change to a degree two Galois extension L . A locally free sheaf on X is called absolutely split if it splits after base change as a direct sum of invertible sheaves on the projective space. We classify the isomorphism classes of absolutely split locally free sheaves on X .

Moduli of ADHM sheaves and the local Donaldson–Thomas theory

The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve X. In particular it is proven that this moduli space is virtually smooth and related by relative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipande and Thomas. This yields a new conjectural construction for the local Donaldson–Thomas theory of curves as well as a natural higher rank generalization.

Matrix factorizations over projective schemes

We study matrix factorizations of locally free coherent sheaves on a scheme. For a scheme that is projective over an affine scheme, we show that homomorphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we give another proof of Orlov's theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. We also give a complete description of the image of this functor.

Inflectional loci of scrolls over smooth, projective varieties

Let X subset of P-N be a scroll over an m-dimensional variety Y. We find the locally free sheaves on X governing the osculating behavior of X, and, under certain dimension assumptions, we compute the cohomology class and the degree of the inflectional locus of X. The case m = 1 was treated in [9]. Here we treat the case m >= 2, which is more complicated for at least two reasons: the expression for the osculating sheaves and the computations of the class of the inflectional locus become more complex, and the dimension requirements needed to ensure validity of the formulas are more severe.

The isogeny conjecture for A-motives

We prove the isogeny conjecture for A-motives over finitely generated fields K of transcendence degree ≤1. This conjecture says that for any semisimple A-motive M over K, there exist only finitely many isomorphism classes of A-motives M′ over K for which there exists a separable isogeny M′→M. The result is in precise analogy to known results for abelian varieties and for Drinfeld modules and will have strong consequences for the ${\mathfrak {p}}$ -adic and adelic Galois representations associated to M. The method makes essential use of the Harder–Narasimhan filtration for locally free coherent sheaves on an algebraic curve.

On the modular interpretation of the Nagaraj–Seshadri locus

We construct a moduli space for torsion free sheaves 𝒜 of rank r and degree zero on an irreducible nodal curve X over the complex numbers which are endowed with a homomorphism 𝔡∶⋀r 𝒜 → 𝒪X which is an isomorphism away from the node. It is a degeneration of the moduli space of SLr(ℂ)-bundles on a smooth curve. In many cases, this moduli space puts a scheme structure on the “Nagaraj–Seshadri locus” inside the moduli space of semistable torsion free sheaves of rank r and degree zero.

Examples of Stability of Tensor Products in Positive Characteristic

Let X be projective smooth variety over an algebraically closed field k and let ℰ, ℱ be μ-semistable locally free sheaves on X. When the base field is ℂ, using transcendental methods, one can prove that the tensor product is always a μ-semistable sheaf. However, this theorem is no longer true over positive characteristic; for an analogous theorem one needs the hypothesis of strong μ-semistability; nevertheless, this hypothesis is not a necessary condition. The objective of this paper is to construct, without the strongly μ-semistability hypothesis, a family of locally free sheaves with μ-stable tensor product.

On degeneration of the surface in the fitting compactification of moduli of stable vector bundles

A new compactification of the moduli scheme of Gieseker-stable vector bundles with given Hilbert polynomial on a smooth projective polarized surface (S, H) over a field $$k = \bar k$$ of zero characteristic was constructed in previous papers by the author. Families of locally free sheaves on the surface S are completed by the locally free sheaves on the schemes which are certain modifications of S. We describe the class of modified surfaces that appear in the construction.

Moduli spaces of stable bundles on Calabi–Yau varieties and Donaldson–Thomas invariants

Let Y be a smooth Calabi–Yau hypersurface of P 1 × P where P stands for a P d -bundle over P 1 . We will prove that for many ample line bundles L and certain Chern characters c , the moduli space M ¯ L ( c ) (resp. M L ( c ) ) of L -Gieseker semistable (resp. L -stable ) rank two torsion free sheaves (resp. vector bundles) on Y with Chern character c are smooth and irreducible and we will compute its dimension. Moreover, we will prove that both moduli spaces coincide. As a byproduct of the geometrical description of these moduli spaces, we will compute the Donaldson–Thomas invariants of some Calabi–Yau 3-folds.

Yangians and cohomology rings of Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL n. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspon- dences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of sln(s ±1 , t)) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of coho- mology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space Mn,d of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is

Fixed point loci of moduli spaces of sheaves on toric varieties

Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves of any dimension on an arbitrary nonsingular toric variety X. Using geometric invariant theory (GIT), this allows us to construct explicit moduli spaces of pure equivariant sheaves on X corepresenting natural moduli functors (similar to work of Payne in the case of equivariant vector bundles). The action of the algebraic torus on X lifts to the moduli space of all Gieseker stable sheaves on X and we express its fixed point locus explicitly in terms of moduli spaces of pure equivariant sheaves on X. One of the problems arising is to find an equivariant line bundle on the side of the GIT problem, which precisely recovers Gieseker stability. In the case of torsion free equivariant sheaves, we can always construct such equivariant line bundles. As a by-product, we get a combinatorial description of the fixed point locus of the moduli space of μ-stable reflexive sheaves on X. As an application, we show in a sequel Kool (2009) [25] how these methods can be used to compute generating functions of Euler characteristics of moduli spaces of μ-stable torsion free sheaves on nonsingular complete toric surfaces.

On a new compactification of moduli of vector bundles on a surface. III: Functorial approach

A new compactification for the scheme of moduli for Gieseker-stable vector bundles with prescribed Hilbert polynomial on the smooth projective polarized surface is constructed. We work over the field of characteristic zero. Families of locally free sheaves on the surface are completed with locally free sheaves on schemes which are modifications of . The Gieseker-Maruyama moduli space has a birational morphism onto the new moduli space. We propose the functor for families of pairs 'polarized scheme-vector bundle' with moduli space of such type.Bibliography: 16 titles.

Generalized Divisors and Total Reflexivity

The notion of generalized divisors on schemes is introduced by Hartshorne. It is shown that there exists a bijection between the set of all generalized divisors on a scheme X and the set of all reflexive coherent 𝒪 X -modules which are locally free of rank one at generic points. This bijection, corresponds Cartier divisors to the set of all locally free sheaves of rank one. Our aim in this article is to study the class of generalized divisors that maps to totally reflexive coherent 𝒪 X -modules, under this correspondence. We investigate this class of divisors, that will be called Gorenstein divisors, both over schemes and also over commutative noetherian rings. We show that this class of divisors has usual properties and fits well in the hierarchy of divisors that already exists in the literature.

Quiver flag varieties and multigraded linear series

This paper introduces a class of smooth projective varieties that generalize and share many properties with partial flag varieties of type A. The quiver flag variety Mϑ(Q,r̲) of a finite acyclic quiver Q (with a unique source) and a dimension vector r̲ is a fine moduli space of stable representations of Q. Quiver flag varieties are Mori dream spaces, they are obtained via a tower of Grassmann bundles, and their bounded derived category of coherent sheaves is generated by a tilting bundle. We define the multigraded linear series of a weakly exceptional sequence of locally free sheaves E̲=(OX,E1,…,Eρ) on a projective scheme X to be the quiver flag variety |E̲|:=Mϑ(Q,r̲) of a pair (Q,r̲) encoded by E̲. When each Ei is globally generated, we obtain a morphism ϕ|E̲|:X→|E̲|, realizing each Ei as the pullback of a tautological bundle. As an application, we introduce the multigraded Plücker embedding of a quiver flag variety.

On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics

We decompose each moduli space of semi-stable sheaves on the complex projective plane with support of dimension one and degree four into locally closed subvarieties, each subvariety being the good or geometric quotient of a set of morphisms of locally free sheaves modulo a reductive or a non-reductive group. We find locally free resolutions of length one for all these sheaves and describe them.