This article is the first in a series of planned articles where noncommutative toric varieties are defined and studied. The background of the project can be said to be the noncommutative geometry of braided monoidal categories, starting with noncommutative deformations of an algebraic torus and using this to deform every toric variety on which the torus acts. It is essential that this does not change the combinatorial fan data describing the toric variety.
The article aims to provide physical interpretations of enumerative invariants of toric varieties. The computations of Donaldson-Thomas invariants of a toric threefold \(X\) can be reduced to the problem of locally enumerating noncommutative instantons on each open patch of \(X\), and then glue this local data to a global quantity. This works because noncommutative deformations of \(\mathbb C^3\) are simple enough to construct instantons. However, commutative toric geometry rules must be used to glue together the quantities constructed locally by noncommutative methods. This article contains a precise definition of a noncommutative toric variety, giving a global noncommutative geometry together with a construction of instantons. This article gives a systematic development of the general machinery needed.
The authors points out the motivation for the construction coming from string theory. Mathematically, this system is described by a holonomic \(\mathcal D\)-module. The category of \(\mathcal D\)-modules is in correspondence with the category of modules on a noncommutative variety. A simple example is the correspondence between right ideals of the algebra of differential operators on the affine line and line bundles over a certain noncommutative deformation of \(\mathbb{CP}^2\), and the classification of bundles on noncommutative \(\mathbb{CP}^2\) is related to the construction of instantons on the noncommutative \(\mathbb R^4\).
The general construction given in this article produces new examples of noncommutative varieties, in particular, considering noncommutative deformations of projective toric varieties, new examples of noncommutative Grassmannians are given, and generalizing to flag varieties.
The article review the needed algebraic constructions: The Hopf cocycle twisting procedure, allowing the construction of the deformations in a braided categorical framework. The twisting procedure is applied to define a noncommutative deformation of the complex algebraic torus \((\mathbb C^\times)^n\), extending the standard noncommutative torus and the basic building block for all the constructions in the article. A twist deformation of the algebraic group \(\text{GL}(n)\) is constructed. This construction requires a notion of quantum determinant, which is then described.
The authors compute the related braided exterior algebras of noncommutative minors. This is used to describe the noncommutative geometry of Grassmannians and the flag variety.
The noncommutative algebraic torus is used to give a general definition of noncommutative toric varieties, given by their combinatorial fan data. It is only the algebras of characters that are deformed and not their group structure. Thus the toric varieties are described by the same fan data.
With these combinatorial, local data, categories of quasi-coherent sheaves on generic noncommutative varieties can be constructed. Many of the standard (in the commutative situation) properties are established. Sheaves of differential forms are built. This is of course crucial for the enumeration of instantons following this text.
The particular example of deforming projective toric varieties are made explicit. Then the noncommutative Grassmannians and flag varieties can be made explicit too, as noncommutative quadrics in projective space, through deformations of Plücker embeddings.
Finally, explicit properties of the categories of quasi-coherent sheaves on the noncommutative projective varieties are given. Then also tautological bundles and sheaves of differential forms on the noncommutative Grassmannians follow.
There are several attempts to construct noncommutative algebraic geometry. This article mixes the different attempts in a very cleaver way. Said (extremely) roughly, the most accepted attempt is to let the noncommutative projective variety be the derived torsion category of coherent sheaves. In this article, a covering of categories of this kind is given to a certain class of toric varieties, replacing the function rings. And this works beautiful in this situation. Thus the article illustrates a nice technique, in addition to the solution of the particular problems.