Chow rings of toric varieties defined by atomic lattices

We study a graded algebra over defined by a finite lattice and a subset in , a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice , as the Chow ring of a smooth toric variety that we construct from and . We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Gröbner basis of the relation ideal of D and a monomial basis of D.     

The versal deformation of an isolated toric Gorenstein singularity

Given a lattice polytope Q ⊆ ℝ ⁿ , we define an affine scheme that reflects the possibilities of splitting Q into a Minkowski sum. Denoting by Y the toric Gorenstein singularity induced by Q, we construct a flat family over with Y as special fiber. In case Y has an isolated singularity, this family is versal. Oblatum 9-V-1996 ⇐p; 30-IX-1996 This paper was written at M.I.T. and supported by a DAAD-fellowship This article was processed by the author using the LAT_EX style file pljour 1m from Springer-Verlag.     

Rationality for generic toric rings

We study generic toric rings. We prove that they are Golod rings, so the Poincaré series of the residue field is rational. We classify when such a ring is Koszul, and compute its rate. Also resolutions related to the initial ideal of the toric ideal with respect to reverse lexicographic order are described. Received August 13, 1997; in final form October 23, 1998     

The Chow ring of punctual Hilbert schemes on toric surfaces

Let X be a smooth projective toric surface, and the Hilbert scheme parametrizing the length d zero-dimensional subschemes of X. We compute the rational Chow ring . More precisely, if is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.     

The multiplicative group action on singular varieties and Chow varieties

We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A brief survey on the structure of Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's problem on the rationality of the irreducible components of Chow varieties.     

The $${\overline{\partial}}$$-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L ^p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.      

Complete toric varieties with reductive automorphism group

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.     

Equivariant-constructible Koszul duality for dual toric varieties

For affine toric varieties X and defined by dual cones, we define an equivalence of categories between mixed versions of the equivariant derived category and the derived category of sheaves on which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel.     

Modularity of abelian varieties over Q with bad reduction in one prime only

We show that certain abelian varieties over Q with bad reduction at one prime only are modular by using methods based on the tables of Odlyzko and class field theory.     

K-duality for pseudomanifolds with an isolated singularity

We associate to a pseudomanifold X with an isolated singularity a differentiable groupoid G which plays the role of the tangent space of X. We construct a Dirac element D and a Dual Dirac element λ which induce a Poincaré duality in K-theory between the $\text{C}{}^1$-algebras C(X) and $\text{C}{}^1\text{(G)}$. To cite this article: C. Debord, J.-M. Lescure, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Darboux-Jouanolou-Ghys integrability for one-dimensional foliations on toric varieties

We use the existence of homogeneous coordinates for simplicial toric varieties to prove a result analogous to the Darboux-Jouanolou-Ghys integrability theorem for the existence of rational first integrals for one-dimensional foliations.     

Uniform K-stability for extremal metrics on toric varieties

In this paper we prove that for toric varieties the uniform K-stability is a necessary condition for the existence of extremal metrics.     

On the resolution of singularities in affine toric 3- varieties

Let $x_{\Sigma(\sigma)}=\ {\rm spec C[\check \sigma \cap Z}^{n}]$ be an affine toric variety given by the monoid algebra $\rm C[\check \sigma \cap Z^{n}]$ , $\check \sigma$ the negative dual cone of a lattice cone σ ? Rⁿ, Σ(σ) the fan consisting of the faces of σ. Assume XΣ(σ) to have only quotient singularities. For n = 3 we classify all pairs X_Σ′, X_Σ(σ) which occur in minimal models of equivariant resolutions Φ: X_Σ′ → - X_Σ(σ) sucn that the regular toric variety XΣ′ has Picard number at most 3.     

Derived category of toric varieties with small Picard number

This paper aims to construct a full strongly exceptional collection of line bundles in the derived category D ^b (X), where X is the blow up of ? ^n?r ×? ^r along a multilinear subspace ? ^n?r?1×? ^r?1 of codimension 2 of ? ^n?r ×? ^r . As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.