Deformation Quantization of Algebraic Varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.     

Embedding and partial resolution of complex cones over Fano threefolds

This work deals with the study of embeddings of toric Calabi–Yau fourfolds which are complex cones over the smooth Fano threefolds. In particular, we focus on finding various embeddings of Fano threefolds inside other Fano threefolds and study the partial resolution of the latter in hope to find new toric dualities.     

Virasoro constraints and the Chern classes of the Hodge bundle

We analyse the consequences of the Virasoro conjecture of Eguchi, Hori and Xiong for Gromov-Witten invariants, in the case of zero degree maps to the manifolds and (or more generally, smooth projective curves and smooth simply connected projective surfaces). We obtain predictions involving intersections of psi and lambda classes on . In particular, we show that the Virasoro conjecture for implies the numerical part of Faber's conjecture on the tautological Chow ring of M_g.     

Mirror Map as Generating Function of Intersection Numbers: Toric Manifolds with Two Kähler Forms

In this paper, we extend our geometrical derivation of the expansion coefficients of mirror maps by localization computation to the case of toric manifolds with two Kähler forms. In particular, we consider Hirzebruch surfaces F ₀, F ₃ and Calabi-Yau hypersurface in weighted projective space P(1, 1, 2, 2, 2) as examples. We expect that our results can be easily generalized to arbitrary toric manifolds.     

On the AKSZ Formulation of the Poisson Sigma Model

We review and extend the Alexandrov–Kontsevich–Schwarz–Zaboronsky construction of solutions of the Batalin–Vilkovisky classical master equation. In particular, we study the case of sigma models on manifolds with boundary. We show that a special case of this construction yields the Batalin–Vilkovisky action functional of the Poisson sigma model on a disk. As we have shown in a previous paper, the perturbative quantization of this model is related to Kontsevich's deformation quantization of Poisson manifolds and to his formality theorem. We also discuss the action of diffeomorphisms of the target manifolds.     

Local structure of Jacobi–Nijenhuis manifolds

After a brief review on the basic notions and the principal results concerning the Jacobi manifolds, the relationship between homogeneous Poisson manifolds and conformal Jacobi manifolds, and also the compatible Jacobi manifolds, we give a generalization of some of these results needed for the contents of this paper. We introduce the notion of Jacobi–Nijenhuis structure and we study the relation between Jacobi–Nijenhuis manifolds and homogeneous Poisson–Nijenhuis manifolds. We present a local classification of homogeneous Poisson–Nijenhuis manifolds and we establish some local models of Jacobi–Nijenhuis manifolds.     

Classical BV Theories on Manifolds with Boundary

In this paper we extend the classical BV framework to gauge theories on spacetime manifolds with boundary. In particular, we connect the BV construction in the bulk with the BFV construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners. We present several examples including electrodynamics, Yang-Mills theory and topological field theories coming from the AKSZ construction, in particular, the Chern-Simons theory, the BF theory, and the Poisson sigma model. This paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing.     

Differential Graded Contact Geometry and Jacobi Structures

We study contact structures on nonnegatively graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization.     

Completing the web of ℤ3‐quotients of complete intersection Calabi‐Yau manifolds

We complete the study [1] of smooth ℤ₃‐quotients of complete intersection Calabi‐Yau threefolds by discussing the six new manifolds that admit free ℤ₃ actions that were discovered in [2]. These manifolds were missed in [1] and complete the web of smooth ℤ₃‐quotients in a nice way. We discuss the transitions between these manifolds and include also the other manifolds of the web. This leads to the conclusion that the web of ℤ₃‐free quotients of complete intersection Calabi‐Yau threefolds is connected by conifold transitions.     

Picard–Fuchs Uniformization and Modularity¶of the Mirror Map

Arithmetic properties of mirror symmetry (type IIA-IIB string duality) are studied. We give criteria for the mirror map q-series of certain families of Calabi–Yau manifolds to be automorphic functions. For families of elliptic curves and lattice polarized K3 surfaces with surjective period mappings, global Torelli theorems allow one to present these criteria in terms of the ramification behavior of natural algebraic invariants – the functional and generalized functional invariants respectively. In particular, when applied to one parameter families of rank 19 lattice polarized K3 surfaces, our criterion demystifies the Mirror-Moonshine phenomenon of Lian and Yau and highlights its non-monstrous nature. The lack of global Torelli theorems and presence of instanton corrections makes Calabi–Yau threefold families more complicated. Via the constraints of special geometry, the Picard–Fuchs equations for one parameter families of Calabi–Yau threefolds imply a differential equation criterion for automorphicity of the mirror map in terms of the Yukawa coupling. In the absence of instanton corrections, the projective periods map to a twisted cubic space curve. A hierarchy of “algebraic” instanton corrections correlated with the differential Galois group of the Picard–Fuchs equation is proposed. Received: 14 August 1999 / Accepted: 30 January 2000     

Gromov-Witten classes,quantum cohomology,and enumerative geometry

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given.     

Poisson manifolds and their associated stacks

We associate to any integrable Poisson manifold a stack, i.e., a category fibered in groupoids over a site. The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. We show that two Poisson manifolds are symplectically Morita equivalent if and only if their associated stacks are isomorphic. We also discuss the non-integrable case.     

Calabi-Yau hypersurfaces in products of semi-ample surfaces

We study Calabi-Yau manifolds that are embedded as hypersurfaces in products of semi-ample complex surfaces. We classify the deformation classes of the latter and thereby achieve a classification of the Calabi-Yau manifolds that are constructed in this way. Complementing the results in the existing literature, we obtain the complete Hodge diamond for all Calabi-Yau hypersurfaces in products of semi-ample surfaces.     

Weyl tensor classification in four-dimensional manifolds of all signatures

It is well known that the classification of the Weyl tensor in Lorentzian manifolds of dimension four, the so called Petrov classification, was a great tool to the development of general relativity. Using the bivector approach it is shown in this article a classification for the Weyl tensor in all four-dimensional manifolds, including all signatures and the complex case, in an unified and simple way. The important Petrov classification then emerges just as a particular case in this scheme. The boost weight classification is also extended here to all signatures as well to complex manifolds. For the Weyl tensor in four dimensions it is established that this last approach produces a classification equivalent to the one generated by the bivector method.     

Separation of variables for integrable systems on Poisson manifolds

For an integrable system on Poisson manifolds, a construction of separated variables is discussed. We suppose that, for a given integrable system, we know a realization of the corresponding Lagrangian submanifold as the product of plane curves. In this case, we can use properties of the foliation of the initial Poisson manifold on symplectic leaves and values of the Casimir functions in order to construct separated variables.     

Poisson Quasi-Nijenhuis Manifolds

We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one correspondence with symplectic (quasi)-Nijenhuis groupoids. As an application, we study generalized complex structures in terms of Poisson quasi-Nijenhuis manifolds. We prove that a generalized complex manifold corresponds to a special class of Poisson quasi-Nijenhuis structures. As a consequence, we show that a generalized complex structure integrates to a symplectic quasi-Nijenhuis groupoid, recovering a theorem of Crainic. Francqui fellow of the Belgian American Educational Foundation. Research supported by NSF grant DMS03-06665 and NSA grant 03G-142.     

Fedosov’s formal symplectic groupoids and contravariant connections

Using Fedosov’s approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kähler–Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variables. We show that the dual of a semisimple Lie algebra does not admit torsion-free Poisson contravariant connections.     

Holomorphic last multipliers on complex manifolds

The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic last multipliers with real last multipliers. Also, we consider some applications of our study for holomorphic gradient vector fields on holomorphic Riemannain manifolds as well as for holomorphic Hamiltonian vector fields and holomorphic Poisson bivector fields on holomorphic Poisson manifolds.