Ann Glob Anal Geom https://doi.org/10.1007/s10455-018-9616-2 The adapted hyper-Kähler structure on the crown domain 1 1 Laura Geatti · 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 and ad 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 1 Introduction A hyper-complex structure on a 4n-dimensional real manifold consists of three complex structures I, J, K such that IJ K =−Id. It is called hyper-Kähler if there exist 2-forms ω , ω , ω which are Kähler for I, J, K , respectively, and deﬁne the same Riemannian I J K metric given by g( · , · ) = ω ( · , I · ) = ω ( · , J ·) = ω
Annals of Global Analysis and Geometry – Springer Journals
Published: Jun 5, 2018
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