Lev A. Borisov

The orbifold Chow ring of toric Deligne-Mumford stacks

Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.

Mirror duality and string-theoretic Hodge numbers

Abstract. We prove in full generality the mirror duality conjecture for string-theoretic Hodge numbers of Calabi–Yau complete intersections in Gorenstein toric Fano varieties. The proof is based on properties of intersection cohomology.

SYSTEMATIC APPROACH TO CYCLIC ORBIFOLDS

We introduce an orbifold induction procedure which provides a systematic construction of cyclic orbifolds, including their twisted sectors. The procedure gives counterparts in the orbifold theory of all the current-algebraic constructions of conformal field theory and enables us to find the orbifold characters and their modular transformation properties.

Elliptic genera of toric varieties and applications to mirror symmetry

Abstract.The paper contains a proof that elliptic genus of a Calabi-Yau manifold is a Jacobi form, finds in which dimensions the elliptic genus is determined by the Hodge numbers and shows that elliptic genera of a Calabi-Yau hypersurface in a toric variety and its mirror coincide up to sign. The proof of the mirror property is based on the extension of elliptic genus to Calabi-Yau hypersurfaces in toric varieties with Gorenstein singularities.

Elliptic genera of singular varieties

The notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced, and the relation between them is studied. The elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties introduced earlier. The orbifold elliptic genus is given in terms of the fixed-point sets of the action. We show that the generating function for the orbifold elliptic genus ∑ Ellorb(X, 6n)p for symmetric groups 6n acting on n-fold products coincides with the one proposed by R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde. The two notions of elliptic genera are conjectured to coincide.

Vertex Algebras and Mirror Symmetry

Abstract: Mirror Symmetry for Calabi–Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi–Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi–Yau manifolds. This should eventually allow us to rewrite the whole story of toric Mirror Symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties.This paper could also be of interest to physicists, because it contains explicit description of holomorphic parts of A and B models of Calabi–Yau hypersurfaces and complete intersections in terms of free bosons and fermions.

String cohomology of Calabi¿Yau hypersurfaces via mirror symmetry

Abstract We propose a construction of string cohomology spaces for Calabi–Yau hypersurfaces that arise in Batyrevs mirror symmetry construction. The spaces are defined explicitly in terms of the corresponding reflexive polyhedra in a mirror-symmetric manner. We draw connections with other approaches to the string cohomology, in particular with the work of Chen and Ruan.

The class of the affine line is a zero divisor in the Grothendieck ring

We show that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. The argument is based on the Pfaffian-Grassmannian double mirror correspondence.

The Pfaffian-Grassmannian derived equivalence

We argue that there exists a derived equivalence between Calabi-Yau threefolds obtained by taking dual linear sections (of the appropriate codimension) of the Grassmannian G(2, 7) and the Pfaffian Pf(7). The existence of such an equivalence has been conjectured by physicists for almost ten years, as the two families of Calabi-Yau threefolds are believed to have the same mirror. It is the first example of a derived equivalence between Calabi-Yau threefolds which are provably non-birational.

The Origin Recognition Complex Interacts with a Subset of Metabolic Genes Tightly Linked to Origins of Replication

The origin recognition complex (ORC) marks chromosomal sites as replication origins and is essential for replication initiation. In yeast, ORC also binds to DNA elements called silencers, where its primary function is to recruit silent information regulator (SIR) proteins to establish transcriptional silencing. Indeed, silencers function poorly as chromosomal origins. Several genetic, molecular, and biochemical studies of HMR-E have led to a model proposing that when ORC becomes limiting in the cell (such as in the orc2-1 mutant) only sites that bind ORC tightly (such as HMR-E) remain fully occupied by ORC, while lower affinity sites, including many origins, lose ORC occupancy. Since HMR-E possessed a unique non-replication function, we reasoned that other tight sites might reveal novel functions for ORC on chromosomes. Therefore, we comprehensively determined ORC “affinity” genome-wide by performing an ORC ChIP–on–chip in ORC2 and orc2-1 strains. Here we describe a novel group of orc2-1–resistant ORC–interacting chromosomal sites (ORF–ORC sites) that did not function as replication origins or silencers. Instead, ORF–ORC sites were comprised of protein-coding regions of highly transcribed metabolic genes. In contrast to the ORC–silencer paradigm, transcriptional activation promoted ORC association with these genes. Remarkably, ORF–ORC genes were enriched in proximity to origins of replication and, in several instances, were transcriptionally regulated by these origins. Taken together, these results suggest a surprising connection among ORC, replication origins, and cellular metabolism.