Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry

Alesker, Semyon; Verbitsky, Misha

doi:10.1007/bf02922058

Cited by 52 publications

(87 citation statements)

References 14 publications

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“…(The ∂∂ J -lemma cf. [4]) A form Ω ∈ Ω 2,0 I,R (M ) satisfies ∂Ω = 0 if and only if it can be locally represented as Ω = ∂∂ J f for a function f ∈ C ∞ (M, R).…”

Section: Hkt Manifoldsmentioning

confidence: 99%

On a uniform estimate for the quaternionic Calabi problem

Alesker

Shelukhin

2013

Isr. J. Math.

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“…(The ∂∂ J -lemma cf. [4]) A form Ω ∈ Ω 2,0 I,R (M ) satisfies ∂Ω = 0 if and only if it can be locally represented as Ω = ∂∂ J f for a function f ∈ C ∞ (M, R).…”

Section: Hkt Manifoldsmentioning

confidence: 99%

On a uniform estimate for the quaternionic Calabi problem

Alesker

Shelukhin

2013

Isr. J. Math.

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“…For such manifolds, quaternionic plurisubharmonic functions can be defined, and a version of the Aleksandrov and Chern-Levine-Nirenberg theorems can be proved. Next, it turns out that the C ∞ -smooth strictly plurisubharmonic functions on hypercomplex manifolds admit a geometric interpretation as (local) potentials of HKT-metrics; this was also shown in [8]. Roughly, an HKT-metric on a hypercomplex manifold is an SU (2)-invariant Riemannian metric satisfying certain first order differential equations.…”

Section: Examplementioning

confidence: 82%

“…In §6 we describe generalizations of some of the definitions and results on quaternionic plurisubharmonic functions to the so-called hypercomplex manifolds; these generalizations were obtained by Verbitsky and the author in [8]. This class of manifolds contains, for instance, the flat spaces H n and the hyper-Kähler manifolds.…”

Section: Examplementioning

confidence: 99%

“…§6. Generalizations to hypercomplex manifolds Some of the definitions and results on quaternionic plurisubharmonic functions discussed above were extended by Verbitsky and the author [8] to a more general context of the so-called hypercomplex manifolds. In this section we give an overview of these results, including a geometric interpretation of the quaternionic strictly plurisubharmonic functions as (local) potentials of HKT-metrics.…”

Section: 5mentioning

confidence: 99%

“…In this section we give an overview of these results, including a geometric interpretation of the quaternionic strictly plurisubharmonic functions as (local) potentials of HKT-metrics. The exposition follows [8].…”

Section: 5mentioning

confidence: 99%

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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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On the Alesker-Verbitsky Conjecture on HyperKähler Manifolds

Dinew

Sroka

2023

Geom. Funct. Anal.

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We solve the quaternionic Monge–Ampère equation on hyperKähler manifolds. In this way we prove the ansatz for the conjecture raised by Alesker and Verbitsky claiming that this equation should be solvable on any hyperKähler with torsion manifold, at least when the canonical bundle is trivial holomorphically. The novelty in our approach is that we do not assume any flatness of the underlying hypercomplex structure which was the case in all the approaches for the higher order a priori estimates so far. The resulting Calabi–Yau type theorem for HKT metrics is discussed.

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Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry

Cited by 52 publications

References 14 publications

On a uniform estimate for the quaternionic Calabi problem

On a uniform estimate for the quaternionic Calabi problem

Untitled

On the Alesker-Verbitsky Conjecture on HyperKähler Manifolds

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