A Counterexample for Subadditivity of Multiplier Ideals on Toric Varieties

We construct a 3-dimensional complete intersection toric variety on which the subadditivity formula doesn't hold, answering negatively a question by Takagi and Watanabe. A combinatorial proof of the subadditivity formula on 2-dimensional normal toric varieties is also provided.     

Existence of Indecomposable Rank Two Vector Bundles on Higher Dimensional Toric Varieties

In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ? ⁿ is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ? ⁿ .     

Combinatorics and Quotients of Toric Varieties

Abstract. For linear projections of polytopes and fans of cones we introduce some new objects such as: virtual chambers, virtual cones and (locally) coherent costrings. Virtual chambers (cones) generalize real chambers (cones), while (locally) coherent costrings are linear dual to (locally) coherent strings. We establish various correspondences for these objects and their connections to toric geometry.     

Combinatorics and Quotients of Toric Varieties

Abstract. For linear projections of polytopes and fans of cones we introduce some new objects such as: virtual chambers, virtual cones and (locally) coherent costrings. Virtual chambers (cones) generalize real chambers (cones), while (locally) coherent costrings are linear dual to (locally) coherent strings. We establish various correspondences for these objects and their connections to toric geometry.     

Toric varieties as spectra of homogeneous prime ideals

We describe the construction of a class of toric varieties as spectra of homogeneous prime ideals.      

Bredon-style homology, cohomology and Riemann-Roch for algebraic stacks

One of the main obstacles for proving Riemann-Roch for algebraic stacks is the lack of cohomology and homology theories that are closer to the K-theory and G-theory of algebraic stacks than the traditional cohomology and homology theories for algebraic stacks. In this paper we study in detail a family of cohomology and homology theories which we call Bredon-style theories that are of this type and in the spirit of the classical Bredon cohomology and homology theories defined for the actions of compact topological groups on topological spaces. We establish Riemann-Roch theorems in this setting: it is shown elsewhere that such Riemann-Roch theorems provide a powerful tool for deriving formulae involving virtual fundamental classes associated to dg-stacks, for example, moduli stacks of stable curves provided with a virtual structure sheaf associated to a perfect obstruction theory. We conclude the present paper with a brief application of this nature.     

Betti Numbers of Toric Varieties and Eulerian Polynomials

It is well known that the Eulerian polynomials, which count permutations in S _n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the S _n -orbit for a generic weight in the weight lattice of S _n . Therefore, the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra.

In this article we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a nongeneric weight, namely, a weight fixed by only the simple reflections J = {s _n , s _n?1, s _n?2,…, s _n?k+2, s _n?k+1} for some k with respect to the A _n root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincaré polynomial of X(J) in terms of the Eulerian polynomials.     

On the S 2-Fication of Some Toric Varieties

In this article we prove the following:

1. Some results on the Cohen–Macaulayness of the canonical module;

2. We study the S ₂-fication of rings which are quotients by lattices ideals;

3. Given a simplicial lattice ideal of codimension two I, its Macaulayfication is given explicitly from a system of generators of I.

     

K-Theory of Algebraic Tori and Toric Varieties

It is proved that under certain conditions the group K _n (X) of a smooth projective variety X over a field F is a natural direct summand of K _n (A) for some separable F-algebra A. As an application we study the K-groups of toric models and toric varieties. A presentation in terms of generators and relations of the groupK ₀(T) for an algebraic torus T is given.     

Asymptotically Maximal Families of Hypersurfaces in Toric Varieties

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope Δ there are families of hypersurfaces with the Newton polytopes that are asymptotically maximal when λ tends to infinity. We also show that these results generalize to complete intersections.     

Toric degenerations of flag varieties from matching field tableaux

We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Plücker algebras. We show that each such family of tableaux leads to a toric ideal, that can be realized as initial of the Plücker ideal, hence a toric degeneration for the flag variety.     

Quasi-linear Quotients and Shellability of Pure Simplicial Complexes

For a square-free monomial ideal I ? S = k[x ₁, x ₂,…, x _n ], we introduce the notion of quasi-linear quotients. By using the quasi-linear quotients, we give a new algebraic criterion for the shellability of a pure simplicial complex Δ over [n]. Also, we provide a new criterion for the Cohen–Macaulayness of the face ring of a pure simplicial complex Δ. Moreover, we show that the face ring of the spanning simplicial complex (defined in [2 Anwar , I. , Raza , Z. , Kashif , A. Spanning simplicial complexes of uni-cyclic graphs . To appear in Algebra Colloquium . [Google Scholar]]) of an r-cyclic graph is Cohen–Macaulay.     

On the Chiral Ring of Calabi–Yau Hypersurfaces in Toric Varieties

This paper is devoted to the calculation of the B-model chiral ring, used in physics, for semiample Calabi–Yau hypersurfaces. We also study the cohomology of semiample hypersurfaces.     

Toric degenerations of Gelfand-Cetlin systems and potential functions

We define a toric degeneration of an integrable system on a projective manifold, and prove the existence of a toric degeneration of the Gelfand-Cetlin system on the flag manifold of type A. As an application, we calculate the potential function for a Lagrangian torus fiber of the Gelfand-Cetlin system.     

Higher K-Theory of Toric Varieties

A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both Quillen's fundamental result on K-homotopy invariance of regular rings and the stable version of the triviality of vector bundles on affine toric varieties. Moreover, it yields a similar behavior of not necessarily affine toric varieties and, further, of their equivariant closed subsets. The conjecture is equivalent to the claim that the relevant admissible morphisms of the category of vector bundles on an affine toric variety can be supported by monomials not in a nondegenerate corner subcone of the underlying polyhedral cone. We prove that one can in fact eliminate all lattice points in such a subcone, except maybe one point. The elimination of the last point is also possible in 0 characteristic if the action of the big Witt vectors satisfies a very natural condition. A partial result of this in the arithmetic case provides first nonsimplicial examples, actually an explicit infinite series of combinatorially different affine toric varieties, simultaneously verifying the conjecture for all higher groups.Supported by the Deutsche Forschungsgemeinschaft, INTAS grant 99-00817 and TMR grant ERB FMRX CT-97-0107     

Function Spaces and Continuous Algebraic Pairings for Varieties

Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including (1) the existence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle theory; (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product; and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.     

Generalizations of Strassen's Equations for Secant Varieties of Segre Varieties

In this article, we compute the center of the infinitesimal Hecke algebras H _z associated to 𝔰𝔩₂; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category 𝒪. We also discuss central elements in infinitesimal Hecke algebras over 𝔤𝔩 _n and 𝔰𝔭(2n) for all n. We end by proving an analogue of Duflo's theorem for H _z .