17 may 2018 -- 11:50
Cortona
Abstract.
Locally conformally Kaehler (LCK) metrics are conformal generalizations of Kaehler metrics. One can consider the fundamental form of such a metric and adapt to this context most of the things one does in symplectic geometry. In particular, there exists a natural notion of Hamiltonian action of a Lie group with respect to it, and so we can speak of toric LCK manifolds.
I will introduce and motivate these notions, along with some examples. I will then give a sketch of the proof that any compact toric LCK manifold admits a Vaisman metric. This is a particularly well behaved metric, whose existence translates into important features of the geometry of the manifold. Finally, I will explain how this, together with other known results, can lead to a classification of toric LCK manifolds.