Coherent Sheaves Research Articles

Fourier-Mukai partners of K3 surfaces in positive characteristic

We study Fourier-Mukai equivalence of K3 surfaces in positive characteristic and show that the classical results over the complex numbers all generalize. The key result is a positive-characteristic version of the Torelli theorem that uses the derived category in place of the Hodge structure on singular cohomology; this is proven by algebraizing formal lifts of Fourier-Mukai kernels to characteristic zero. As a consequence, any Shioda-supersingular K3 surface is uniquely determined up to isomorphism by its derived category of coherent sheaves. We also study different realizations of Mukai's Hodge structure in algebraic cohomology theories (etale, crystalline, de Rham) and use these to prove: 1) the zeta function of a K3 surface is a derived invariant (discovered independently by Huybrechts); 2) the variational crystalline Hodge conjecture holds for correspondences arising from Fourier-Mukai kernels on products of two K3 surfaces.

Геометрические свойства коммутативных подалгебр дифференциальных операторов в частных производных

We investigate further alebro-geometric properties of commutative rings of partial differential operators continuing our research started in previous articles. In particular, we start to explore the most evident examples and also certain known examples of algebraically integrable quantum completely integrable systems from the point of view of a recent generalization of Sato's theory which belongs to the second author. We give a complete characterisation of the spectral data for a class of "trivial" rings and strengthen geometric properties known earlier for a class of known examples. We also define a kind of a restriction map from the moduli space of coherent sheaves with fixed Hilbert polynomial on a surface to analogous moduli space on a divisor (both the surface and divisor are part of the spectral data). We give several explicit examples of spectral data and corresponding rings of commuting (completed) operators, producing as a by-product interesting examples of surfaces that are not isomorphic to spectral surfaces of any commutative ring of PDOs of rank one. At last, we prove that any commutative ring of PDOs, whose normalisation is isomorphic to the ring of polynomials $k[u,t]$, is a Darboux transformation of a ring of operators with constant coefficients.

Stability Conditions for Slodowy Slices and Real Variations of Stability

We provide examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau, and propose a new variant of definition of stabilities on a triangulated category, which we call a "real variation of stability conditions". We discuss its relation to Bridgeland's definition; the main theorem provides an illustration of such a relation. We also state a conjecture by the second author and Okounkov relating this structure to quantum cohomology of symplectic resolutions and establish its validity in some special cases. More precisely, let X be the standard resolution of a transversal slice to an adjoint nilpotent orbit of a simple Lie algebra over C. An action of the affine braid group on the derived category of coherent sheaves on X and a collection of t-structures on this category permuted by the action have been constructed in arXiv:1101.3702 and arXiv:1001.2562 respectively. In this note we show that the t-structures come from points in a certain connected submanifold in the space of Bridgeland stability conditions. The submanifold is a covering of a submanifold in the dual space to the Grothendieck group, and the affine braid group acts by deck transformations. In the special case when dim (X)=2 a similar (in fact, stronger) result was obtained in arXiv:math/0508257.

Absolute purity in the category of quasi coherent sheaves

In this paper we consider the class of absolutely pure and Fp-injective quasi-coherent sheaves. We show that these two classes of quasi-coherent sheaves over a locally coherent scheme are equivalent. As a corollary we will show that the class of absolutely pure quasi-coherent sheaves over such a scheme is an enveloping and a covering class. It is proved that over a locally coherent scheme, the pair (?(Abs(X),Abs(X)) is a cotorsion theory. The existence of a duality between absolutely pure envelopes and flat covers is proved as expected.

Operational $K$-theory

We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of operational Chow theory. This operational K -theory agrees with Grothendieck groups of vector bundles on smooth varieties, admits a natural map from the Grothen­dieck group of perfect complexes on general varieties, satisfies descent for Chow envelopes, and is {A}^1 -homotopy invariant. Furthermore, we show that the operational K -theory of a complete linear variety is dual to the Grothendieck group of coherent sheaves. As an application, we show that the K -theory of perfect complexes on any complete toric threefold surjects onto this group. Finally we identify the equivariant operational K -theory of an arbitrary toric variety with the ring of integral piecewise exponential functions on the associated fan.

DG categories and exceptional collections

A description of how to assign to a full exceptional collection on a variety X X a DG category C \mathcal {C} such that the bounded derived category of coherent sheaves on X X is equivalent to the bounded derived category of C \mathcal {C} is given in a 1990 work by Bondal and Kapranov, Framed triangulated categories. In this paper we show that the category C \mathcal {C} can be chosen to have finite-dimensional spaces of morphisms. We describe how it behaves under mutations and present an algorithm allowing us to calculate it for full exceptional collections with vanishing Ext k ^k groups for k > 1 k>1 .

Tame–wild dichotomy of Birkhoff type problems for nilpotent linear operators

Inspired by recent results of Ringel–Schmidmeier, Kussin–Lenzing–Meltzer, Xiao-Wu Chen, and Pu Zhang, given a field K, m≥1, and a finite poset I≡(I,⪯) with a unique maximal element ⁎, we study the category Mon(I,Fm) of mono-representations of I over the Frobenius K-algebra Fm:=K[t]/(tm) of K-dimension m<∞, viewed as K-vector spaces U⁎, with an m-nilpotent K-linear operator t:U⁎→U⁎, together with t-invariant subspaces Ui⊆Uj⊆U⁎, for all i⪯j⪯⁎ in I. The problem of when the Krull–Schmidt K-category Mon(I,Fm) is of wild (resp. tame) representation type is called a wild (resp. tame) Birkhoff type problem for m-nilpotent operators. In case when K is algebraically closed, we give a complete solution of the problem by describing all minimal pairs (I,m) (resp. all pairs), with m≥1, such that category Mon(I,Fm) is of wild (resp. tame) representation type. We reduce the problem to a Birkhoff type problem for the category fspr(I•,Fm)⊆Mon(I•,Fm) of subprojective representations over Fm of a larger poset I•⊃I. The tame–wild dichotomy for the category Mon(I,Fm) is also proved.Surprisingly, in case when I=Ia,b is the union of two incomparable chains I′ and I″ of length |I′|=a−1≥1 and |I″|=b−1≥1, with I′∩I″={⁎}, the problem is equivalent with the wildness (resp. tameness) of the category coh-X(p) of coherent sheaves over the weighted projective line X(p), for the weight triple p=(a,b,m), with a,b,m≥2, studied by Kussin, Lenzing and Meltzer [19] in relation with the hypersurface singularity f=x1a+x2b+x3m.

Constructible sheaves on affine Grassmannians and geometry of the dual nilpotent cone

In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group G, contructible with respect to the stratification by \(G({\Bbb C}\left[\kern-0.15em\left[ x \right]\kern-0.15em\right])\)-orbits. Following ideas of Ginzburg and Arkhipov-Bezrukavnikov-Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of the Langlands dual reductive group G. We also show, in the mixed case, that restriction to the nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on affine Grassmannians.

Singular support of coherent sheaves and the geometric Langlands conjecture

We define the notion of singular support of a coherent sheaf on a quasi-smooth-derived scheme or Artin stack, where “quasi-smooth” means that it is a locally complete intersection in the derived sense. This develops the idea of “cohomological” support of coherent sheaves on a locally complete intersection scheme introduced by D. Benson, S. B. Iyengar, and H. Krause. We study the behavior of singular support under the direct and inverse image functors for coherent sheaves. We use the theory of singular support of coherent sheaves to formulate the categorical geometric Langlands conjecture. We verify that it passes natural consistency tests: It is compatible with the geometric Satake equivalence and with the Eisenstein series functors. The latter compatibility is particularly important, as it fails in the original “naive” form of the conjecture.

The derived category of a GIT quotient

Given a quasiprojective algebraic variety with a reductive group action, we describe a relationship between its equivariant derived category and the derived category of its geometric invariant theory quotient. This generalizes classical descriptions of the category of coherent sheaves on projective space and categorifies several results in the theory of Hamiltonian group actions on projective manifolds.This perspective generalizes and provides new insight into examples of derived equivalences between birational varieties. We provide a criterion under which two different GIT quotients are derived equivalent, and apply it to prove that any two generic GIT quotients of an equivariantly Calabi-Yau projective-over-affine manifold by a torus are derived equivalent.

Weighted Projective Lines as Fine Moduli Spaces of Quiver Representations

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their canonical bundles, and show that they are rarely tilting. We also give a moduli construction for these total spaces for weighted projective lines with three orbifold points.

Universal Plane Curve and Moduli Spaces of 1-dimensional Coherent Sheaves

We show that the universal plane curve M of fixed degree d ≥ 3 can be seen as a closed subvariety in a certain Simpson moduli space of 1-dimensional sheaves on ℙ2 contained in the stable locus. The universal singular locus of M coincides with the subvariety M′ of M consisting of sheaves that are not locally free on their support. It turns out that the blow up Bl M′ M may be naturally seen as a compactification of M B = M∖M′ by vector bundles (on support).

Ind-abelian categories and quasi-coherent sheaves

AbstractWe study the question of when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.

Tilting bundles and the missing part on a weighted projective line of type [formula omitted

This paper is devoted to investigate the tilting bundles, i.e. tilting sheaves that are vector bundles, in the category of coherent sheaves on a weighted projective line of type (2,2,n). We classify all the tilting bundles into two classes, one containing the tilting bundles that are consisting of line bundles, and the other one containing tilting bundles with indecomposable summands that are not line bundles. Moreover, for each tilting bundle T (with endomorphism algebra Λ) we prove that the missing part, from the category of coherent sheaves to the category of finitely generated right Λ-modules, carries the structure of an abelian category.

On the Ext-computability of Serre quotient categories

To develop a constructive description of Ext in categories of coherent sheaves over certain schemes, we establish a binatural isomorphism between the Ext-groups in Serre quotient categories A/C and a direct limit of Ext-groups in the ambient Abelian category A. For Ext1 the isomorphism follows if the thick subcategory C⊂A is localizing. For the higher extension groups we need further assumptions on C. With these categories in mind we cannot assume A/C to have enough projectives or injectives and therefore use Yoneda's description of Ext.

Openness of versality via coherent functors

Abstract We give a proof of openness of versality using coherent functors. As an application, we streamline Artin’s criterion for algebraicity of a stack. We also introduce multi-step obstruction theories, employing them to produce obstruction theories for the stack of coherent sheaves, the Quot functor, and spaces of maps in the presence of non-flatness.

An Algebro-geometric Construction of Lower Central Series of Associative Algebras

The lower central series invariants Mk of an associative algebra A are the two-sided ideals generated by k-fold iterated commutators; the Mk provide a filtration of A. We study the relationship between the geometry of X=Spec Aab and the associated graded components Nk of this filtration. We show that the Nk form coherent sheaves on a certain nilpotent thickening of X, and that Zariski localization on X coincides with noncommutative localization of A. Under certain freeness assumptions on A, we give an alternative construction of Nk purely in terms of the geometry of X (and in particular, independent of A). Applying a construction of Kapranov, we exhibit the Nk as natural vector bundles on the category of smooth schemes.

Stability conditions and birational geometry of projective surfaces

AbstractWe show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.

On nodal Enriques surfaces and quartic double solids

We consider the class of singular double coverings $X \to \PP^3$ ramified in the degeneration locus $D$ of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such quartic surface $D$ one can associate an Enriques surface $S$ which is the factor of the blowup of $D$ by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface $S$ is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of $X$.

A remark on braid group actions on coherent sheaves

A construction of braid group actions on coherent sheaves using mixed Hodge modules and some well known constructions from geometric representation theory is given.