# The geometric structure of symplectic and horospherical contraction (Jeremy Lane, UniGe)

06.10.2017 14:00 – 16:00

Degenerations and their gradient-Hamiltonian flows are a major theme in recent studies of interactions between algebraic geometry, representation theory, and symplectic topology. Although it can be difficult to precisely describe the gradient-Hamiltonian flow of a given degeneration, an improved understanding the geometry of this flow is desirable, since it often leads to interesting new results lying at the interface between algebraic and symplectic topology (cf. papers by Nishinou-Nohara-Ueda, Kaveh, Fang-Littelmann-Pabiniak). Recent work by Hilgert-Manon-Martens (HMM) provides an algebraic formula for the time-1 flow of the degeneration of a semi-projective variety equipped with an action by a connected complex reductive group G to its horospherical contraction. More generally, HMM define the symplectic contraction of any Hamiltonian K-manifold (K compact) and prove that in the algebraic setting, it coincides with horospherical contraction. After an introduction to degenerations and horospherical contraction, I will show that the symplectic contraction of a Hamiltonian K-manifold has a very natural interpretation as a quotient of the manifold by the null foliation of a decomposition into coisotropic submanifolds determined by the action of K and its moment map. This perspective should have applications to the study of Gelfand-Zeiltin integrable systems, which HMM have shown can be constructed by an iterated symplectic contraction.

### Lieu

Bâtiment: Battelle

Séminaire "Groupes de Lie et espaces des modules"

### Organisé par

Section de mathématiques### Intervenants

Jeremy Lane, UniGeentrée libre

### Classement

Catégorie: Séminaire