Ann Glob Anal Geom
The adapted hyper-Kähler structure on the crown
· Andrea Iannuzzi
Received: 7 April 2018 / Accepted: 16 May 2018
© Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract Let be the crown domain associated with a non-compact irreducible Hermitian
symmetric space G/K . We give an explicit description of the unique G-invariant adapted
hyper-Kähler structure on , i.e., compatible with the adapted complex structure J
with the G-invariant Kähler structure of G/K . We also compute invariant potentials of the
involved Kähler metrics and the associated moment maps.
Keywords Hyper-Kähler manifold · Hermitian symmetric space · Crown domain
Mathematics Subject Classiﬁcation 53C26 · 32M15 · 37J15
A hyper-complex structure on a 4n-dimensional real manifold consists of three complex
structures I, J, K such that IJK =−Id. It is called hyper-Kähler if there exist 2-forms
which are Kähler for I, J, K , respectively, and deﬁne the same Riemannian
metric given by
g( · , · ) = ω
( · , I · ) = ω
( · , J ·) = ω
( · , K ·).
A hyper-Kähler manifold is holomorphic symplectic with respect to any of its complex
structures, e.g., the complex symplectic form ω
is holomorphic with respect to I .
Research partially supported by INdAM-GNSAGA.
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientiﬁca 1, 00133