Potentials for hyper-Kähler metrics with torsion

Banos, Bertrand; Swann, Andrew

doi:10.1088/0264-9381/21/13/004

Cited by 29 publications

(39 citation statements)

References 11 publications

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“…Consider the map of vector bundles t : Λ 2,0 I,R (X) → S H (X) defined by t(η)(A, A) = η(A, A • J) for any (real) vector field A on X. Then t is an isomorphism of vector bundles (this was proved in [39] (ii) The proof of Theorem 6.13 employs a result of Banos-Swann [11].…”

Section: 12mentioning

confidence: 99%

Untitled

2008

St. Petersburg Math. J.

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Abstract. The paper is a survey of the recent theory of plurisubharmonic functions of quaternionic variables, together with its applications to the theory of valuations on convex sets and HKT-geometry (Hyper-Kähler with Torsion). The exposition follows some earlier papers by the author and a joint paper by Verbitsky and the author. §0. IntroductionOur goal in this article is to present a survey of the recent theory of plurisubharmonic functions of quaternionic variables, together with its applications to the theory of valuations on convex sets and HKT-geometry (Hyper-Kähler with Torsion). The exposition follows the papers [4,5,7] by the author and [8] by Verbitsky and the author.We denote by H the (noncommutative) field of quaternions. The notion of a quaternionic plurisubharmonic function on the flat space H n was introduced by the author in [4] and independently by Henkin in [24]. This notion is a quaternionic analog of a convex function on R n and a complex plurisubharmonic function on C n ; see Definition 3.1 below. On the one hand, this class of functions enjoys many analytic properties similar to those of convex and complex plurisubharmonic functions. On the other hand, these properties reflect rather different geometric structures behind them. This will be illustrated below by applications to convexity and HKT-geometry.We start with some analytic properties of quaternionic plurisubharmonic functions. In [4], the author proved a quaternionic analog of the Aleksandrov [2] and Chern-LevineNirenberg [19] theorems (see Theorems 3.4 and 3.6 in §3 below). It is worthwhile to recall these classical results now. The Aleksandrov theorem says that if a sequence {f N

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Section: 12mentioning

confidence: 99%

Untitled

2008

St. Petersburg Math. J.

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“…Consider the map of vector bundles t : Λ (ii) The proof of Theorem 6.13 employs a result of Banos-Swann [11].…”

Section: 12mentioning

confidence: 99%

Quaternionic plurisubharmonic functions and their applications to convexity

Alesker

2007

St. Petersburg Math. J.

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“…We show that any HPKT-structure is locally generated by a real (potential) function following the ideas developed in [4]. To this end, using Salamon's idea from the quaternionic case (see [30]), we define an invariant first order differential operator D, the hyperparacomplex operator, on an almost hyper-paracomplex manifold and we show that it is 2-step nilpotent exactly when the almost hyper-paracomplex structure is integrable.…”

Section: Introductionmentioning

confidence: 95%

“…We provide hyper-parahermitian versions of many local and some global results for hyper-hermitian manifolds, specially we adopt the hyper-complex constructions of [16,14,4] (but see also [30,24,25,33]). …”

Section: Introductionmentioning

confidence: 99%

Hyper-parahermitian manifolds with torsion

Ivanov

Tsanov

Zamkovoy

2006

Journal of Geometry and Physics

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Necessary and sufficient conditions for the existence of a hyper-parahermitian connection with totally skew-symmetric torsion (HPKT-structure) are presented. It is shown that any HPKT-structure is locally generated by a real (potential) function. An invariant first order differential operator is defined on any almost hyper-paracomplex manifold showing that it is two-step nilpotent exactly when the almost hyper-paracomplex structure is integrable. A local HPKT-potential is expressed in terms of this operator. Examples of (locally) invariant HPKT-structures with closed as well as non-closed torsion 3-form on a class of (locally) homogeneous hyper-paracomplex manifolds (some of them compact) are constructed.

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Potentials for hyper-Kähler metrics with torsion

Abstract: We prove that locally any hyper-Kähler metric with torsion admits an HKT potential.

Cited by 29 publications

References 11 publications

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Quaternionic plurisubharmonic functions and their applications to convexity

Hyper-parahermitian manifolds with torsion

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