# The adapted hyper-Kaehler structure on the tangent bundle of a Hermitian symmetric space II

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Laura Geatti

created by risa on 16 Feb 2018

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.