Hypercomplex structures on Kähler manifolds
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In this paper, a compact Kahler manifold admitting a hypercomplex structure (M, I, J, K) admits a natural HKT-metric, which is used to construct a holomorphic symplectic form on (m, I).Abstract:
Let (M, I) be a compact Kahler manifold admitting a hypercomplex structure (M, I, J, K). We show that (M, I, J, K) admits a natural HKT-metric. This is used to construct a holomorphic symplectic form on (M, I).read more
Citations
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Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds
Semyon Alesker,Misha Verbitsky +1 more
TL;DR: A quaternionic version of the Calabi problem on the Monge-Ampere equation is introduced in this paper, which is a special case of the complex Hessian equation, making sense on any complex manifold.
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Hypercomplex manifolds with trivial canonical bundle and their holonomy
TL;DR: In this paper, the holonomy of Obata connection on compact nilmanifolds equipped with abelian hypercomplex structures was shown to be contained in SL(n,H) by showing that the canonical bundle of (M,I,J,K) is trivial as a holomorphic line bundle.
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Existence of HKT metrics on hypercomplex manifolds of real dimension 8
TL;DR: In this paper, it was shown that a hypercomplex manifold M with the Obata holonomy contained in S L ( 2, H ) admits a hyperkahler with torsion (HKT) structure if and only if H 1 ( O (M, I, K ) is even-dimensional.
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Some remarks on Calabi–Yau and hyper-Kähler foliations
Georges Habib,Luigi Vezzoni +1 more
TL;DR: In this article, the authors studied Riemannian foliations whose transverse Levi-Civita connection ∇ has special holonomy and showed that a simply-connected compact manifold with a Kahler foliation admits a transverse hyper-Kahler structure if and only if it admits a compatible transverse Hyper-Hermitian structure.
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Some remarks on Calabi-Yau and hyper-K\"ahler foliations
Georges Habib,Luigi Vezzoni +1 more
TL;DR: In this paper, the authors studied Riemannian foliations whose transverse Levi-Civita connection has special holonomy and showed that a simply-connected compact manifold with a Kahler foliation admits a transverse hyper- Kahler structure if and only if it admits a compatible transverse Hermitian structure.
References
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Book
Principles of Algebraic Geometry
Phillip Griffiths,Joe Harris +1 more
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
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On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*
TL;DR: In this paper, the Ricci form of some Kahler metric is shown to be closed and its cohomology class must represent the first Chern class of M. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.
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Twistor spaces for hyper-Kähler manifolds with torsion
P.S. Howe,G. Papadopoulos +1 more
TL;DR: In this paper, the authors construct the twistor space associated with a hyper-Kahler manifold with torsion, a type of geometry that arises as the target space geometry in two-dimensional sigma models with (4,0) supersymmetry.