Northwestern, Notre Dame, UIC Complex Geometry Seminar

 Abstract: This talk will be about symplectic, rather than complex geometry, but the motivation is from constructions in K\”ahler geometry. We consider a compact symplectic manifold $(X,\omega)$ and the group $G$ of its symplectomorphisms. We study the action of $G$ on the Fr\’echet space $C^\infty(X)$ of smooth functions, by pullback, and describe properties of convex functions $p:C^\infty(X)\to\mathbb R$ that are invariant under this action.

 Abstract: This talk will be about symplectic, rather than complex geometry, but the motivation is from constructions in K\”ahler geometry. We consider a compact symplectic manifold $(X,\omega)$ and the group $G$ of its symplectomorphisms. We study the action of $G$ on the Fr\’echet space $C^\infty(X)$ of smooth functions, by pullback, and describe properties of convex functions $p:C^\infty(X)\to\mathbb R$ that are invariant under this action.

When constructing canonical metrics in complex geometry, it is useful to consider potentials of metrics with less regularity. For example, T. Darvas showed that the completion of the space of Kahler potentials with respect to a natural metric is the space of potentials of finite energy. I will discuss joint work with S. Boucksom, where we study a trivially valued analogue of this picture, with test configurations playing the role of Kahler metrics.