Quantum deformations of projective three-space
type: Item
instance of: scholarly article
title: Quantum deformations of projective three-space
zbMATH Open document ID: 1338.14007
author: Brent Pym
published in: Advances in Mathematics
publication date: 2015-07-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1403.6444
review text: The complex projective space \(\mathbb P^n\) has no deformations as a classical algebraic variety, but several noncommutative ones which the author calls quantum deformations. The article concentrates on the case \(n=3\), and relations to problems in Poisson geometry and foliation theory. Noncommutative projective geometry is modelled by the homogeneous coordinate rings, noncommutative analogues of polynomial rings satisfying the Artin-Schelter (AS) regularity criterion. From the fact that a Poisson structure on a surface is determined by a section of the anticanonical bundle, there is a link between the noncommutative algebras and the geometry of anticanonical divisors in the classical surfaces. The case of \(\mathbb P^3\) in the form of AS regular algebras of global dimension 4 has been studied extensively, and the full classification remains one of the central questions in noncommutative projective geometry. From the subclass of AS regular algebras consisting of the Calabi-Yau algebras, it is possible to obtain other AS regular algebras by twisting procedures. Although a given quantum deformation of \(\mathbb P^n\) may be representable by many different AS regular algebras, there will be a unique one that is Calabi-Yau. The main result of the article is a classification of the quantum deformations of \(\mathbb P^3\): The flat deformations the graded Calabi-Yau algebra \(\mathbb C[x_0,x_1,x_2,x_3]\) is contained in six irreducible families, each realized as closures of the \(\mathrm{GL}(4,\mathbb C)\)-orbits of specified normal forms. The families intersect nontrivially, and it is not quite clear how to describe all possible intersections. Thus the focus is on suitable generic elements of each family. To prove the main result, the problem of classification is reduced to the study of the semi-classical limits using Kontsevich's results on deformation quantization. Such deformation quantization of a given Poisson structure will be a Calabi-Yau algebra if and only if…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 37F75
zbMATH DE Number: 6458150
zbMATH Keywords: noncommutative projective geometry
zbMATH Keywords: Calabi-Yau algebra
zbMATH Keywords: Possin structure
zbMATH Keywords: holomorphic foliation
zbMATH Keywords: noncommutative deformation
zbMATH Keywords: foliations
zbMATH Keywords: deformation quantization
zbMATH Keywords: quadratic algebra
zbMATH Keywords: skew polynomial ring
zbMATH Keywords: quantum deformations
zbMATH Keywords: Kozul algbebra
zbMATH Keywords: superpotensial
describes a project that uses: SageMath
describes a project that uses: GAP
describes a project that uses: surfex
MaRDI profile type: Publication
OpenAlex ID: W2963671549
arXiv ID: 1403.6444
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DOI: 10.1016/J.AIM.2015.06.005