NDLI: Hermitian structures on six-dimensional nilmanifolds

Content Provider	Springer Nature Link
Author	Ugarte, Luis
Copyright Year	2007
Abstract	Let (J,g) be a Hermitian structure on a six-dimensional compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We show that, up to equivalence of the complex structure, the strong Kahler with torsion structures (J,g) on M are parametrized by the points in a subset of the Euclidean space, in particular, the region inside a certain ovaloid corresponds to such structures on the Iwasawa manifold and the region outside to strong Kahler with torsion structures with nonabelian J on the nilmanifold $\Gamma\backslash (H^3\times H^3),$ where H3 is the Heisenberg group. A classification of six-dimensional nilmanifolds admitting balanced Hermitian structures (J,g) is given, and as an application we classify the nilmanifolds having invariant complex structures which do not admit Hermitian structure with restricted holonomy of the Bismut connection contained in SU(3). It is also shown that on the nilmanifold $\Gamma\backslash (H^3\times H^3)$ the balanced condition is not stable under small deformations. Finally, we prove that a compact quotient of $H(2,1)\times \mathbb{R},$ where H(2,1) is the five-dimensional generalized Heisenberg group, is the only six-dimensional nilmanifold having locally conformal Kahler metrics, and the complex structures underlying such metrics are all equivalent. Moreover, this nilmanifold is a Vaisman manifold for any invariant locally conformal Kahler metric.
Starting Page	175
Ending Page	202
Page Count	28
File Format	PDF
ISSN	10834362
Journal	Transformation Groups
Volume Number	12
Issue Number	1
e-ISSN	1531586X
Language	English
Publisher	Birkhäuser-Verlag
Publisher Date	2007-02-14
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Topological Groups, Lie Groups Algebra
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Geometry and Topology

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