28/06/2018, 16:30 — 17:30 — Room P4.35, Mathematics Building
José Mourão, CAMGSD, Instituto Superior Técnico, Universidade de Lisboa
Imaginary time Hamiltonian flows and applications to Kahler geometry, Kahler reduction and representation theory
The formalism to complexify time in the flow of a nonholomorphic vector field on a complex manifold is reviewed. The complexified flow, besides acting on $M$, changes also the complex structure. We will describe the following applications:
- For a compact Kahler manifold the imaginary time Hamiltonian flows correspond to Mabuchi geodesics in the infinite dimensional space of Kahler metrics on $M$. These geodesics play a very important role in the study of stability of Kahler manifolds. A nontrivial nontoric example on the two-dimensional sphere will be described.
- Let the compact connected Lie group $G$ act in an Hamiltonian and Kahler way on a Kahler manifold $M$ and assume that its action extends to $G_C$. Then, by taking geodesics of Kahler structures generated by convex functions of the $G$-momentum to infinite geodesic time, one gets (conjecturally always, proved on several important examples) a concentration of holomorphic sections of holomorphic line bundles on inverse images of coadjoint orbits under the $G$-momentum map. A nontrivial toric example and the case of $M=G_C$ will be described.
On work with T Baier, J Hilgert, O Kaya, JP Nunes, M Pereira, P Silva.