Dans les années 60, Vinberg a décrit comment à partir d'un polyèdre projectif muni de réflexions projectives, on pouvait construire un ouvert convexe $\Omega$ sur lequel le groupe de Coxeter $W$ associé au polyèdre agissait. Vinberg a donné une CNS pour que l'action de $W$ sur $\Omega$ soit cocompact. Je donnerai une CS pour que l'action de $W$ sur $\Omega$ soit de covolume fini, ou convexe-cocompact.
By "adding degenerations", the moduli spaces of Kähler-Einstein (K-polystable) Fano manifolds should admit the structure of a projective variety. In this talk, I will explain the expected picture, focusing on the understood two dimensional case of Del Pezzo surfaces (joint work with Yuji Odaka and Song Sun).
We show that pseudo-holomorphic polygons in a Liouville-domain can be lifted to the symplectization of its contactization.
In particular, Legendrian contact homology may equivalently be defined by counting either of these objects. We use this
fact to prove an isomorphism between the linearized Legendrian contact homology induced by an exact Lagrangian filling and
the singular homology of the filling, a result which was first conjectured by Seidel.
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