Toric degenerations of spherical varieties

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.     

Toric sheaves,stability and fibrations

For an equivariant reflexive sheaf over a polarised toric variety, we study slope stability of its reflexive pullback along a toric fibration. Examples of such fibrations include equivariant blow-ups and toric locally trivial fibrations. We show that stability (resp. unstability) is preserved under such pullbacks for so-called adiabatic polarisations. In the strictly semistable situation, under locally freeness assumptions, we provide a necessary and sufficient condition on the graded object to ensure stability of the pulled back sheaf. As applications, we provide various stable perturbations of semistable tangent sheaves, either by changing the polarisation, or by blowing-up a subvariety. Finally, our results apply uniformly in specific flat families and induce injective maps between the associated moduli spaces.     

Differential operators on toric varieties and Fourier transform

We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point of the affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of .     

Toric sets and orbits on toric varieties

Let D be an integer matrix. A toric set, namely the points in Kⁿ parametrized by the columns of D, and a toric variety are associated to D. The toric set is a subset of the toric variety. We describe the relation between the toric set and the toric variety, in terms of the orbits of the torus action on the toric variety. The toric set depends on the sign (+,−,0) pattern of the matrix D. Finally, we prove that any toric variety over an algebraically closed field can be expressed as a toric set, for an appropriate matrix.     

Local Euler obstructions of toric varieties

We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones.     

On Euler–Jaczewski sequence and Remmert–Van de Ven problem for toric varieties

Let X be a complete toric variety and Y a smooth projective variety with . We prove that, if is a surjective morphism then . Received: 15 May 2001; in final form: 22 October 2001/ Published online: 4 April 2002     

Abelian surfaces in projective toric 4-folds

We show fundamental properties on embedding of abelian surfaces into projective toric 4-folds, and study the case of the toric Del Pezzo 4-fold from the viewpoint of the moduli space of abelian surfaces with polarization of type (1, 5). Received: 31 January 2005; revised: 15 April 2005     

𝔾a-ACTIONS ON AFFINE CONES

An affine algebraic variety X is called cylindrical if it contains a principal Zariski dense open cylinder U ? Z × A1. A polarized projective variety (Y, H) is called cylindrical if it contains a cylinder U = Y \ supp D, where D is an effective Q-divisor on Y such that [D] ∈ Q+[H] in PicQ(Y ). We show that cylindricity of a polarized projective variety is equivalent to that of a certain Veronese affine cone over this variety. This gives a criterion of the existence of a unipotent group action on an affine cone.     

Moduli stacks of stable toric quasimaps

We construct new “virtually smooth” modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental's compactifications, when the complex structure of the curve is allowed to vary and markings are included, and are the toric counterpart of the moduli spaces of stable quotients introduced by Marian, Oprea, and Pandharipande to compactify spaces of maps to Grassmannians. A brief discussion of the resulting invariants and their (conjectural) relation with Gromov-Witten theory is also included.     

Fiber Fans and Toric Quotients

The GIT chamber decomposition arising from a subtorus action on a polarized quasiprojective toric variety is a polyhedral complex. Denote by Σ the fan that is the cone over the polyhedral complex. In this paper we show that the toric variety defined by the fan Σ is the normalization of the toric Chow quotient of a closely related affine toric variety by a complementary torus.     

On toric varieties which are almost set-theoretic complete intersections

We describe a class of affine toric varieties V that are set-theoretically minimally defined by binomial equations over fields of any characteristic.     

Arithmetic toric varieties

We study toric varieties over a field k that split in a Galois extension using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k‐forms of projective spaces when is cyclic, and we also study k‐forms of surfaces.     

Toric P-difference varieties

In this paper, we introduce the concept of P-difference varieties and study the properties of toric P-difference varieties. Toric P-difference varieties are analogues of toric varieties in difference algebraic geometry. The category of affine toric P-difference varieties with toric morphisms is shown to be antiequivalent to the category of affine P [x]-semimodules with P [x]-semimodule morphisms. Moreover, there is a one-to-one correspondence between the irreducible invariant P-difference subvarieties of an affine toric P-difference variety and the faces of the corresponding affine P [x]-semimodule. We also define abstract toric P-difference varieties by gluing affine toric P-difference varieties. The irreducible invariant P-difference subvariety-face correspondence is generalized to abstract toric P-difference varieties. By virtue of this correspondence, a divisor theory for abstract toric P-difference varieties is developed.     

Secant varieties of toric varieties

Let X_P be a smooth projective toric variety of dimension n embedded in P^r using all of the lattice points of the polytope P. We compute the dimension and degree of the secant variety . We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties X_A embedded using a set of lattice points A⊂P∩Zⁿ containing the vertices of P and their nearest neighbors.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998