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Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties

Robert J. Berman, Bo Berndtsson (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in ℝ n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P . Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties ( X , Δ ) saying that ( X , Δ ) admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and...

Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Fabrizio Catanese, Frédéric Mangolte (2009)

Annales scientifiques de l'École Normale Supérieure

Let W → X be a real smooth projective 3-fold fibred by rational curves such that W ( ℝ ) is orientable. J. Kollár proved that a connected component N of W ( ℝ ) is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when X is a geometrically rational surface. When N is Seifert fibred over a base orbifold F , our result generalizes...

Representation stability for syzygies of line bundles on Segre–Veronese varieties

Claudiu Raicu (2016)

Journal of the European Mathematical Society

The rational homology groups of packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre–Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability,...

Représentations des algèbres de rang 2 et fonctions zêta associées

Dehbia Achab (1995)

Annales de l'institut Fourier

Dans un travail précédent nous avons défini et étudié la fonction zêta associée à une représentation d’une algèbre de Jordan euclidienne déployée et à un réseau dans l’espace de la représentation. Nous avons démontré la convergence dans un demi-plan, établi l’existence d’un prolongement méromorphe et d’une équation fonctionnelle scalaire. Cette fonction est une généralisation de la fonction zêta de Koecher; elle est donnée dans son domaine de convergence, par une série qui somme sur certains éléments...

Representations of the Kauffman bracket skein algebra of the punctured torus

Jea-Pil Cho, Răzvan Gelca (2014)

Fundamenta Mathematicae

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.

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