相关论文: Deformation Quantization in Algebraic Geometry

In this paper, we study quantization on a compact integral symplectic manifold $X$ with transversal real polarizations. In the case of complex polarizations, namely $X$ is K\"ahler equipped with transversal complex polarizations $T^{1, 0}X,…

Coisotropic deformations of algebraic varieties are defined as those for which an ideal of the deformed variety is a Poisson ideal. It is shown that coisotropic deformations of sets of intersection points of plane quadrics, cubics and space…

Let $\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}$ be a variational Poisson bracket in a field model on an affine bundle $\pi$ over an affine base manifold $M^m$. Denote by $\times$ the commutative associative multiplication in the…

Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop some technique for computing cohomology of such graph complexes and apply it to several concrete…

A method for the deformation quantization of coadjoint orbits of semisimple Lie groups is proposed. It is based on the algebraic structure of the orbit. Its relation to geometric quantization and differentiable deformations is explored.

First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation…

We give an elementary proof of the result by Leichtnam, Tang, and Weinstein that there exists a deformation quantization with separation of variables on a complex manifold endowed with a Kaehler-Poisson structure vanishing on a Levi…

We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. $P_n$-algebras. Roughly speaking, a coisotropic morphism is given by a $P_{n+1}$-algebra acting on a…

We consider a smooth Poisson affine variety with the trivial canonical bundle over complex numbers. For such a variety the deformation quantization algebra A_h enjoys the conditions of the Van den Bergh duality theorem and the corresponding…

This paper investigates the geometry of a smooth canonically polarized surface $X$ defined over an algebraically closed field of characteristic $p>0$ in the case when the automorphism scheme of $X$ is not smooth. This is a situation that…

In this paper we prove formality of the exterior algebra on V+V* endowed with the big bracket considered as a graded Poisson algebra. We also discuss connection of this result to bialgebra deformations of the symmetric algebra of V…

We consider antiPoisson superalgebra realized on the smooth Grassmann-valued functions of the form \xi f_0(x)+f_1(x), where f_0 has compact support on R, and with the parity opposite to that of the Grassmann superalgebra realized on these…

Realizing a part of the Derived Deformation Theory program, we construct a "derived" analog of the Grothendieck's Quot scheme parametrizing subsheaves in a given coherent sheaf F on a smooth projective variety X. This analog is a…

We introduce a general theory of twisting algebraic structures based on actions of a bialgebra. These twists are closely related to algebraic deformations and also to the theory of quasi-triangular bialgebras. In particular, a deformation…

For a ringed space (X,O), we show that the deformations of the abelian category Mod(O) of sheaves of O-modules are obtained from algebroid prestacks, as introduced by Kontsevich. In case X is a quasi-compact separated scheme the same is…

As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain non-linear Poisson brackets which are ``cocycle perturbations'' of the linear Poisson bracket. We show that these special Poisson…

Since a Poisson Structure is a smooth bivector field, we use a ring-valued sheaf $\OO_{X}$ on a manifold with corners $X$, we can interpret $\OO_{X}(U)$ as the ring of admissible smooth functions where $U$ is an open subset on $X$, in this…

The deformation theory of singular varieties plays a central role in understanding the geometry and moduli of algebraic varieties. For a variety $X$ with possibly singular points, the space of first-order infinitesimal deformations is given…

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective or embeddable into toric varieties. Our methods…

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.