Rational intersection cohomology of projective toric varieties.
Rational points and Coxeter group actions on the cohomology of toric varieties
We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.
Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties
We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in with exponential non-linearity and target a convex body is solvable iff is the barycenter of Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties saying that admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and...
Representation of the Hirzebruch-Kleinschmidt varieties by quadrics.
Representations of the Kauffman bracket skein algebra of the punctured torus
We describe the action of the Kauffman bracket skein algebra on some vector spaces that arise as relative Kauffman bracket skein modules of tangles in the punctured torus. We show how this action determines the Reshetikhin-Turaev representation of the punctured torus. We rephrase our results to describe the quantum group quantization of the moduli space of flat SU(2)-connections on the punctured torus with fixed trace of the holonomy around the boundary.
Riemann sums over polytopes
It is well-known that the -th Riemann sum of a compactly supported function on the real line converges to the Riemann integral at a much faster rate than the standard rate of convergence if the sum is over the lattice, . In this paper we prove an n-dimensional version of this result for Riemann sums over polytopes.
Scalar product of Dirichlet series and the distribution of integer points on toric varieties.
Simple conditions on conic varieties.
Singular foliations of toric type
Stabilization of monomial maps in higher codimension
A monomial self-map on a complex toric variety is said to be -stable if the action induced on the -cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of , we can find a toric model with at worst quotient singularities where is -stable. If is replaced by an iterate one can find a -stable model as soon as the dynamical degrees of satisfy . On the other hand, we give examples of monomial maps , where this condition...
Stably rational algebraic tori
The rationality of a stably rational torus with a cyclic splitting field is proved.
Stratification theory from the Newton polyhedron point of view
Recently, T. Fukui and L. Paunescu introduced a weighted version of the -regularity condition and Kuo’s ratio test condition. In this approach, we consider the - regularity condition and -regularity related to a Newton filtration.
Sur des variétés toriques non projectives
Sur la géométrie des variétés sphériques.
Tame homomorphisms of polytopal rings.
The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings
In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding . In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions...
The classification of weighted projective spaces
We obtain two classifications of weighted projective spaces: up to hoeomorphism and up to homotopy equivalence. We show that the former coincides with Al Amrani's classification up to isomorphism of algebraic varieties, and deduce the latter by proving that the Mislin genus of any weighted projective space is rigid.
The cohomology ring of polygon spaces
We compute the integer cohomology rings of the “polygon spaces”introduced in [F. Kirwan, Cohomology rings of moduli spaces of vector bundles over Riemann surfaces, J. Amer. Math. Soc., 5 (1992), 853-906] and [M. Kapovich & J. Millson, the symplectic geometry of polygons in Euclidean space, J. of Diff. Geometry, 44 (1996), 479-513]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory...
The Denef-Loeser series for toric surface singularities.
Let H denote the set of formal ares going through a singular point of an algebraic variety V defined over an algebraically closed field k of charactcristic zcro. In the late sixties, J, Nash has observed that for any nonnegative integer s, the set js(H) of s-jets of ares in H is a constructible subset of some affine space. Recently (1999), J. Denef and F. Loeser have proved that the Poincaré series associated with the image of js(H) in some suitable localization of the Grothendieck ring of algebraic...
The Ehrhart polynomial of a lattice -simplex.