12 research outputs found

    Strominger connection and pluriclosed metrics

    Full text link
    In this paper, we prove a conjecture raised by Angella, Otal, Ugarte, and Villacampa recently, which states that if the Strominger connection (also known as Bismut connection) of a compact Hermitian manifold is K\"ahler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a K\"ahler manifold, then the metric must be pluriclosed. Actually, we show that Strominger K\"ahler-like is equivalent to the pluriclosedness of the Hermitian metric plus the parallelness of the torsion, even without the compactness assumption

    Complex nilmanifolds and K\"ahler-like connections

    Full text link
    In this note, we analyze the question of when will a complex nilmanifold have K\"ahler-like Strominger (also known as Bismut), Chern, or Riemannian connection, in the sense that the curvature of the connection obeys all the symmetries of that of a K\"ahler metric. We give a classification in the first two cases and a partial description in the third case. It would be interesting to understand these questions for all Lie-Hermitian manifolds, namely, Lie groups equipped with a left invariant complex structure and a compatible left invariant metric

    On local stabilities of pp-K\"ahler structures

    Full text link
    By use of a natural extension map and a power series method, we obtain a local stability theorem for p-K\"ahler structures with the (p,p+1)(p,p+1)-th mild ∂∂ˉ\partial\bar\partial-lemma under small differentiable deformations.Comment: Several typos have been fixed. Final version to appear in Compositio Mathematica. arXiv admin note: text overlap with arXiv:1609.0563

    On Strominger K\"ahler-like manifolds with degenerate torsion

    Full text link
    In this paper, we study a special type of compact Hermitian manifolds that are Strominger K\"ahler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is K\"ahler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a K\"ahler manifold. Previously, we have shown that any SKL manifold (Mn,g)(M^n,g) is always pluriclosed, and when the manifold is compact and gg is not K\"ahler, it can not admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n=2n=2, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-K\"ahler SKL manifolds in dimension 33 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-K\"ahler SKL manifold, its K\"ahler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal K\"ahler when n≥3n\geq 3, and the manifold does not admit any Hermitian symplectic metric
    corecore