Hyperkähler manifolds with torsion, supersymmetry and Hodge theory

Abstract

Abstract. Let M be a hypercomplex Hermitian manifold, (M, /) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linearalgebraic construction produces a canonical nowhere degenerate (2 7 0 . Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics. We exploit a remarkable analogy between the de Rham DG-algebra of a Kahler manifold and the Dolbeault DG-algebra of an HKT-manifold. The supersymmetry of a Kahler manifold X is given by an action of an 8-dimensional Lie superalgebra g on A*(X), containing the Lefschetz £L(2)-triple, the Laplacian and the de Rham differential. We establish the action of # on the Dolbeault DG-algebra A*' 0 (M, /) of an HKT-manifold. This is used to construct a canonical Lefschetz-type 5L(2)-action on the space of harmonic spinors of M.

On a uniform estimate for the quaternionic Calabi problem

“…HKT manifolds were introduced in the physical literature by Howe and Papadopoulos [13]. For the mathematical treatment see Grantcharov-Poon [11] and Verbitsky [17]. The original definition of HKT-metrics in [13] was different but equivalent to Definition 5 (the latter was given in [11]).…”

“…A form ω ∈ Ω 2k,0 I (M ) that satisfies Jω = ω will be called a real (2k,0)-form [17]. The space of real (2k, 0)-forms will be denoted by Ω 2k,0 I,R (M ).…”

“…9 Definition. (The t isomorphism [17]) Given a right vector space V over H, there is an R-linear isomorphism t : Λ 2,0 I,R (V ) → S H (V ) from the space of the real (2, 0)-forms on V to the space of symmetric forms on V that are invariant with respect to the action of the group SU (2) of unit quaternions, given by t(η)(A, A) := η(A, A • J) ( [17]). We shall denote by the same letter the induced isomorphism t :…”

Hodge theory on nearly Kähler manifolds

“…Proof To prove Theorem 3.1, we use essentially the same argument as used in the proof of the conventional Kähler identities in the situation when a coordinate approach does not work; see eg our proof of Kähler identities in HKT-geometry, obtained in [29], and our proof of the Kähler identities in locally conformally hyperkähler geometry, obtained in [30].…”

Untitled

“…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].…”