23 feb 2018 -- 15:30
Aula D'Antoni, Dip.Matematica, Università "Tor Vergata", Roma
Abstract.
The cotangent bundle of a compact Hermitian symmetric space $X=G/K$ (a tubular neighbourhood of the zero section, in the non-compact case) carries a unique $G$-invariant hyper-Kaehler structure compatible with the Kaehler structure of $X$ and the canonical complex symplectic form of $T^*X$. The tangent bundle $TX$, which is isomorphic to $T^*X$, carries a canonical complex structure $J$, the so called 'adapted complex structure', and admits a unique $G$-invariant hyper-Kaehler structure compatible with the Kaehler structure of $X$ and the adapted complex structure $J$. The two hyper-Kaehler structures are related by a $G$-equivariant fiber preserving diffeomorphism of $TX$, as already noticed by Dancer and Szoeke. The fact that the domain of existence of $J$ in $TX$ is biholomorphic to a G-invariant domain in the complex homogeneous space $GC/KC$ allows us to use Lie theoretical tools and moment map techniques to explicitly compute the various quantities of the 'adapetd hyper-Kaehler structure'. This is part of a joint project with Andrea Iannuzzi, and this talk concludes his presentation of February 9.