On Strominger Kähler-like manifolds with degenerate torsion

Abstract

In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold. Previously, we have shown that any SKL manifold (M n , g) is always pluriclosed, and when the manifold is compact and g is not Kähler, it can not admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n = 2, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-Kähler SKL manifolds in dimension 3 and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, for any compact non-Kähler SKL manifold, its Kähler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal Kähler when n ≥ 3, and the manifold does not admit any Hermitian symplectic metric.

On Strominger space forms

“…That is, the only non-Kähler Strominger space forms amongst SKL manifolds are the Strominger flat manifolds. SKL manifolds were studied in [38], [39], and [40], where classification theorem in n = 2 and n = 3 as well as some structural results in general dimensions were obtained. SKL manifolds form an interesting subset of pluriclosed manifolds.…”

“…Then ẽ = e −u e and φ = e u ϕ become unitary frame and dual coframe for g. Under respective unitary frames, the connection matrix and torsion for Chern connection are related by (cf. in the proof of Theorem 3 in [38]): (24) θ = θ + (∂u − ∂u)I, T j ik = e −u {T j ik + u i δ jk − u k δ ji } where u k = e k (u). From this, we get the expression for γ, γ hence (25)…”

“…Let (M n , g) be a Strominger Kähler-like (SKL) manifold with pointwise constant Strominger holomorphic sectional curvature. If g is not Kähler, then by [38] we know that locally there always exist admissible frames, which are unitary frames with ∇ s e n = 0, and R s ijkn = 0 for any 1 ≤ i, j, k ≤ n. The SKL condition implies that R s ijkℓ = R s ijkℓ , and the pointwise constant Strominger holomorphic sectional curvature assumption means that there is a function f on M such that R s ijkℓ = f 2 (δ ij δ kℓ + δ iℓ δ kj ). So by R s nnnn = 0, we get f ≡ 0, hence R s = R s = 0, that is, g is Strominger flat.…”

Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature

“…Denote ∇ c to be the Chern connection of g with θ being the connection matrix and { T i jk } being the components of the torsion forms τ i of ∇ c . Direct calculations from the structure equation (see [8], [27]) give that…”

“…There has been many recent studies on the curvatures of special connections on Hermitian manifolds, see e.g. for the Chern connection ( [3], [4], [7], [16], [17], [20], [22]), the Levi-Civita connection ( [2], [14], [21]), the Strominger connection ( [8], [24], [27], [28]) and the Lichnerowicz connection ( [10], [13], [19]). There are also some recent work on the curvatures of general Gauduchon connections, see [1], [23], [26], [30] etc.…”