Quantum deformations of projective three-space
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 the Poisson structure is unimodular. Then the result can be reduced to proving that the variety parametrizing unimodular quadratic Poisson structures in four dimensions has exactly six irreducible components. The author gives explicit formulas for the generic Poisson brackets in each component. This is possible by using work of \textit{D. Cerveau} and \textit{A. Lins Neto} [Ann. Math. (2) 143, No. 3, 577--612 (1996; Zbl 0855.32015)] regarding codimension-one foliations on \(\mathbb P^3\) and recent refinements of Loray, Pereira, and Touzet [\textit{F. Loray} et al., Math. Nachr. 286, No. 8--9, 921--940 (2013; Zbl 1301.37032)] and the well-known correspondence between homogeneous quadratic Poisson structures on projective space. The author consider quantization as equivariant geometry, and finds that the corresponding quantum \(\mathbb P^3\) contains 3 commutative rational curves; a line, a plane conic, and a twisted cubic. These all correspond to various embeddings of \(\mathbb P^1\) in its symmetric power. There is also a pencil of noncommutative sextic surfaces whose classical limits are the level sets of the \(j\)-invariant, and the Schwarzenberger bundles which served as early examples of indecomposable vector bundles on projective space quantize to give graded bimodules over the \(\mathrm E(3)\) algebra. The article reviews basic facts about deformations of quadratic algebras and their semiclassical limits in noncommutative projective geometry. Calabi-Yau algebras are introduced, superpotentials, their role as unique homogeneous coordinate rings for quantum \(\mathbb P^n\)s, and the connection with the unimodularity condition for Poisson structures. To complete the proof of the main result, the author gives normal forms for the appearing algebras and compare their semiclassical limits with the classification of foliations. A quantum projective space is described by a noncommutative graded algebra \(A=\bigoplus_{k\geq 0}A_k\) which arises as a deformation of the usual product on the polynomial ring. Invariantly, consider a vector space \(V\), and present the symmetric algebra as a quotient \(S^\bullet V=T^\bullet V/(\bigwedge V)\) of the tensor algebra by the ideal generated by the skew- symmetric two-tensors. Quantizations of the homogeneous coordinate ring are constructed by deforming the subspace \(\bigwedge^2 V\subset V\otimes V\) to a new element \(R\in\mathrm{Gr}(r,V\otimes V)\) of the relevant Grassmannian. This means that \(A\) is deformed as a quadratic algebra. The deformations are supposed to have the same same invariants as the polynomial ring in the sense that the Hilbert series is constant. This article uses the flat, noncommutative, deformations of the ``commutation ideal'' as the quantum deformations, and these are the main objects of investigation. Define a \textit{superpotential} on a vector space \(V\) of dimension \(d\) as an element \(\Phi\in V^{\otimes d}\) such that \(\mathrm{cyc}(\Phi)=(-1)^d\Phi\), where \(\mathrm{cyc}:V^{\otimes d}\rightarrow V^{\otimes d}\) is the linear automorphism that cyclically permutes the tensor factors. From a (possibly) twisted superpotential, one obtains a quadratic algebra by differentiating, and the link to Calabi-Yau algebras is given by a result stating that A Koszul algebra \(A\) is Calabi-Yau (resp. twisted Calabi-Yau) if and only if it is derived from a superpotential (respectively, twisted superpotential) whose associated complex is exact. The article illustrates the deformation theory in a nice way, containing nice and explicit examples. Its results is important as it illustrates the link between different methods in noncommutative projective geometry.
scientific article