|XXXVIII Workshop on Geometric Methods in Physics||30.06-6.07.2019|
|VIII School on Geometry and Physics||24-28.06.2019|
Imaginary time Hamiltonian flows and applications to Quantization, Kahler geometry and representation theory
The formalism to complexify time in the flow of a vector field on a complex manifold is reviewed. The complexified flow, besides acting on $M$, changes also the complex structure. We will review 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 $M$ that play a very important role in the study of stability of Kahler manifolds. A nontrivial 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 $M$ so that its action extends to $G_C$. Then, by taking geodesics of Kahler structures generated by convex functions of the $G$-moment map 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.
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