Mixed Hodge structures and equivariant sheaves on the projective plane

We describe an equivalence of categories between the category of mixed Hodge structures and a category of equivariant vector bundles on a toric model of the complex projective plane which verify some semistability condition. We then apply this correspondence to define an invariant which generalizes the notion of R ‐split mixed Hodge structure and give calculations for the first group of cohomology of possibly non smooth or non‐complete curves of genus 0 and 1. Finally, we describe some extension groups of mixed Hodge structures in terms of equivariant extensions of coherent sheaves. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Equivariant Chern Classes of Orientable Toric Origami Manifolds∗

A toric origami manifold, introduced by Cannas da Silva, Guillemin and Pires, is a generalization of a toric symplectic manifold. For a toric symplectic manifold, its equivariant Chern classes can be described in terms of the corresponding Delzant polytope and the stabilization of its tangent bundle splits as a direct sum of complex line bundles. But in general a toric origami manifold is not simply connected, so the algebraic topology of a toric origami manifold is more difficult than a toric symplectic manifold. In this paper they give an explicit formula of the equivariant Chern classes of an oriented toric origami manifold in terms of the corresponding origami template. Furthermore, they prove the stabilization of the tangent bundle of an oriented toric origami manifold also splits as a direct sum of complex line bundles.     

The chiral de rham complex and positivity of the equivariant signatures of some loop spaces

In this note we show that the positivity property of the equivariant signature of the loop space, first observed in [MS1] in the case of the even-dimensional projective spaces, is valid for Picard number 2 toric varieties. A new formula for the equivariant signature of the loop space in the case of a toric spin variety is derived.Partially supported by an NSF grant     

Local Gromov-Witten invariants of canonical line bundles of toric surfaces

We define and compute by localizating the local equivariant Gromov-Witten invariants of the canonical line bundles of toric surfaces, not necessarily Fano.     

Riemann-Roch for Quotients and Todd Classes of Simplicial Toric Varieties

Abstract

In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne (Brion M. and Vergne M. ([1997] Brion, M. and Vergne, M. 1997. An equivariant Riemann-Roch theorem for complete simplicial toric varieties. J. Reine. Agnew. Math., 482: 67–92.  [Google Scholar]). An equivariant Riemann-Roch theorem for complete simplicial toric varieties. J. Reine. Agnew. Math.482:67–92) using different techniques.     

Equivariant primary decomposition and toric sheaves

We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of varieties which are quotients of a quasi-affine variety by the action of a diagonalizable group and thus admit a homogeneous coordinate ring, such as toric varieties. Comparing these decompositions with primary decompositions of graded modules over the homogeneous coordinate ring, we show that these are equivalent if the action of the diagonalizable group is free. We give some specific examples for the case of toric varieties.     

Remarks on uniform bundles on projective spaces

On projective spaces of dimension \(d\ge 2\) defined over a field of positive characteristic we construct rank \(d+1\) uniform toric but non-homogeneous bundles, which do not exists in characteristic zero. These bundles are obtained by choosing suitable equivariant extensions of the Frobenius pullbacks of \(T_{\mathbb {P}^d}\) by a line bundle.     

Gluing Affine Torus Actions Via Divisorial Fans

Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a “proper polyhedral divisor” introduced in earlier work, we develop the concept of a “divisorial fan” and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like *-surfaces and projectivizations of (nonsplit) vector bundles over toric varieties.     

Vector bundles on toric varieties

Following Sam Payne?s work, we study the existence problem of nontrivial vector bundles on toric varieties. The first result we prove is that every complete fan admits a nontrivial conewise linear multivalued function. Such functions could potentially be the Chern classes of toric vector bundles. Then we use the results of Cortiñas, Haesemeyer, Walker and Weibel to show that the (non-equivariant) Grothendieck group of the toric 3-fold studied by Payne is large, so the variety has a nontrivial vector bundle. Using the same computation, we show that every toric 3-fold X either has a nontrivial line bundle, or there is a finite surjective toric morphism from Y to X, such that Y has a large Grothendieck group.     

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     

Applications of toric systems on surfaces

In this note we apply the techniques of the toric systems introduced by Hille–Perling to several problems on smooth projective surfaces: We showed that the existence of full exceptional collection of line bundles implies the rationality for small Picard rank surfaces; we proved equivalences of several notions of cyclic strong exceptional collection of line bundles; we also proposed a partial solution to a conjecture on exceptional sheaves on weak del Pezzo surfaces.     

An application of Guillemin-Abreu theory to a non-abelian group action

This note is a step towards demonstrating the benefits of a symplectic approach to studying equivariant Kähler geometry. We apply a local differential geometric framework from Kähler toric geometry due to Guillemin and Abreu to the case of the standard linear SU(n) action on Cn?{0}. Using this framework we (re)construct certain Kähler metrics from data on moment polytopes.     

Constructions in Sasakian geometry

We first generalize the join construction described previously by the first two authors [4] for quasi-regular Sasakian-Einstein orbifolds to the general quasi-regular Sasakian case. This allows for the further construction of specific types of Sasakian structures that are preserved under the join operation, such as positive, negative, or null Sasakian structures, as well as Sasakian-Einstein structures. In particular, we show that there are families of Sasakian-Einstein structures on certain 7-manifolds homeomorphic to S ² × S ⁵. We next show how the join construction emerges as a special case of Lerman’s contact fibre bundle construction [32]. In particular, when both the base and the fiber of the contact fiber bundle are toric we show that the construction yields a new toric Sasakian manifold. Finally, we study toric Sasakian manifolds in dimension 5 and show that any simply-connected compact oriented 5-manifold with vanishing torsion admits regular toric Sasakian structures. This is accomplished by explicitly constructing circle bundles over the equivariant blow-ups of Hirzebruch surfaces. During the preparation of this work the first two authors were partially supported by NSF grants DMS-0203219 and DMS-0504367.     

On a notion of toric special linear systems

In this note, we study linear systems on complete toric varieties X with an invariant point whose orbit under the action of $\mathrm{Aut}(X)$ contains the dense torus T of X. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations.