9 feb 2018 -- 15:30
Aula 1B1, Dip.SBAI, Università "La Sapienza", Roma
Abstract.
In this first talk, we wish to recall basic facts and classical constructions of invariant hyper-Kaehler structures on the cotangent bundle $T^*X$ of a Hermitian symmetric space $X=G/K$ due to Eguchi-Hanson, Calabi, Biquard, Gauduchon, Feix, Kaledin. On (a tubolar neighborhood of) the tangent bundle $TX∼T^*X$ one also has the adapted complex structure J which makes it biholomorphic to the (crown domain in the) complex homogeneous space $GC/KC$. By letting $J$ replace the role of the holomorphic symplectic form on $T^*X$, one obtains the unique adapted hyper-Kaehler structure associated to $G/K$ . In this context, the interplay of complex geometry and the Lie group structure of GC leads to an explicit realization of all the terms of such a structure. Time permitting, we compare the adapted context with the previous constructions. This is part of a joint project with Laura Geatti. More details on the adapted realization in the second talk.