On the unipotence of autoequivalences of toric complete intersection Calabi–Yau categories

Consider the derived category of coherent sheaves, D ^b (X), on a compact Calabi–Yau complete intersection X in a toric variety. The scope of this work is to establish the (quasi-)unipotence of a class of elements in the group of autoequivalences, Aut(D ^b (X)). This is achieved by associating singularity categories, modelled by matrix factorizations, to the toric data. Each of these triangulated categories is equivalent to the derived category of coherent sheaves on X. The idea is then that, although the singularity categories share the group of autoequivalences, on each category there are elements in Aut(D ^b (X)), whose (quasi-)unipotence relations are easier to see than on the other categories.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

On the Action of the Dual Group on the Cohomology of Perverse Sheaves on the Affine Grassmannian

It was proved by Ginzburg, Mirkovic and Vilonen that the G(O)-equivariant perverse sheaves on the affine Grassmannian of a connected reductive group G form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group G . In this paper we construct explicitly the action of G on the global cohomology of a perverse sheaf.     

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U _σ = Spec A ^σ . A quasi-coherent sheaf on X gives rise, by taking sections over the U _σ , to a diagram of modules over the coordinate rings A ^σ , indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Σ^op-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U _σ agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.     

Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

Given a smooth projective toric variety X, we construct an A_∞ category of Lagrangians with boundary on a level set of the Landau–Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X.      

Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described.     

On equivalences of derived and singular categories

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $ \mathbb{A}^1 $ \mathbb{A}^1 , g:Y → $ \mathbb{A}^1 $ \mathbb{A}^1 . Assuming that there exists a complex of sheaves on X × $ \mathbb{A}^1 $ \mathbb{A}^1 Y which induces an equivalence of D ^b (X) and D ^b (Y), we show that there is also an equivalence of the singular derived categories of the fibers f ⁻¹(0) and g ⁻¹(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.     

On the structure of rigid semistable sheaves on algebraic surfaces

LetS be a smooth projective surface, letK be the canonical class ofS and letH be an ample divisor such thatH • K < 0. We prove that for any rigid sheafF (Ext¹ (F, F) = 0) that is Mumford-Takemoto semistable with respect toH there exists an exceptional set (E ₁ ,..., E _n ) of sheaves onS such thatF can be constructed from {E _i } by means of a finite sequence of extensions. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 692–700, November, 1998. The author wishes to express his gratitude to S. A. Kuleshov for useful discussions and to A. N. Rudakov and A. L. Gorodentsev for their attention to the present work. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01323 and by the INTAS Foundation.     

Staggered t-structures on toric varieties

Achar has recently introduced a family of t-structures on the derived category of equivariant coherent sheaves on a G-scheme, generalizing the perverse coherent t-structures of Bezrukavnikov and Deligne. They are called staggered t-structures, and one of their points of interest is that they are more often self-dual. In this paper we investigate these t-structures on the T-equivariant derived category of a toric variety.     

On the classification of toric Fano 4-folds

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ ^d , the so-called Fano polyhedra, up to an isomorphism of the standard lattice . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory Vol. 56. Algebraic Geometry-9, 1998.     

On generators of ideals defining projective toric varieties

 For any ample line bundle L on a projective toric variety of dimension n, it is proved that the line bundle L ^⊗i is normally generated if i is greater than or equal to n−1, and examples showing that this estimate is best possible are given. Moreover we prove an estimate for the degree of the generators of the ideals defining projective toric varieties. In particular, when L is normally generated, the defining ideal of the variety embedded by the global sections of L has generators of degree at most n+1. When the variety is embedded by the global sections of L ^⊗(n−1) , then the defining ideal has generators of degree at most three. Received: 11 July 2001 / Revised version: 17 December 2001     

Residual kernels with singularities on coordinate planes

A finite collection of planes {E _v } in ℂ^d is called an atomic family if the top de Rham cohomology group of its complement is generated by a single element. A closed differential form generating this group is called a residual kernel for the atomic family. We construct new residual kernels in the case when E _v are coordinate planes such that the complement ℂ^d/∪ E _v admits a toric action with the orbit space being homeomorphic to a compact projective toric variety. They generalize the well-known Bochner-Martinelli and Sorani differential forms. The kernels obtained are used to establish a new formula of integral representations for functions holomorphic in Reinhardt polyhedra.     

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X _k obtained by blowing up ℂℙ² at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W _k :M _k →ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X _k , and give an explicit correspondence between the deformation parameters for X _k and the cohomology class [B+iω]∈H ²(M _k ,ℂ).     

Constructions in Sasakian geometry

We first generalize the join construction described previously by the first two authors [4] for quasi-regular Sasakian-Einstein orbifolds to the general quasi-regular Sasakian case. This allows for the further construction of specific types of Sasakian structures that are preserved under the join operation, such as positive, negative, or null Sasakian structures, as well as Sasakian-Einstein structures. In particular, we show that there are families of Sasakian-Einstein structures on certain 7-manifolds homeomorphic to S ² × S ⁵. We next show how the join construction emerges as a special case of Lerman’s contact fibre bundle construction [32]. In particular, when both the base and the fiber of the contact fiber bundle are toric we show that the construction yields a new toric Sasakian manifold. Finally, we study toric Sasakian manifolds in dimension 5 and show that any simply-connected compact oriented 5-manifold with vanishing torsion admits regular toric Sasakian structures. This is accomplished by explicitly constructing circle bundles over the equivariant blow-ups of Hirzebruch surfaces. During the preparation of this work the first two authors were partially supported by NSF grants DMS-0203219 and DMS-0504367.     

The Generalized Auslander–Reiten Condition for the bounded derived category

We consider the bounded derived category D ^b (R mod) of a left Noetherian ring R. We give a version of the Generalized Auslander–Reiten Condition for D ^b (R mod) that is equivalent to the classical statement for the module category and is preserved under derived equivalences.     

The $${{\rm GL}(2,\mathbb{C})}$$ McKay correspondence

In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,\mathbbC){{\rm GL}(2,\mathbb{C})}, it means that the endomorphism ring of the special CM \mathbbC{\mathbb{C}} [[x, y]] ^G -modules can be used to build the dual graph of the minimal resolution of \mathbbC²/G{\mathbb{C}^{2}/G}, extending McKay’s observation (McKay, Proc Symp Pure Math, 37:183–186, 1980) for finite subgroups of SL(2,\mathbbC){{\rm SL}(2,\mathbb{C})} to all finite subgroups of GL(2,\mathbbC){{\rm GL}(2,\mathbb{C})}.     

Linear precision for parametric patches

We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary (and quite restrictive) conditions for a patch to possess linear precision. A main focus is on linear precision for Krasauskas’ toric patches, which we show is equivalent to a certain rational map on \mathbb C\mathbb P^d{\mathbb C}{\mathbb P}^d being a birational isomorphism. Lastly, we establish the connection between linear precision for toric surface patches and maximum likelihood degree for discrete exponential families in algebraic statistics, and show how iterative proportional fitting may be used to compute toric patches.