arXiv subjects
Publications and source records attributed to Quanting Zhao.
We continue our study on Hermitian manifolds that are {\em Bismut torsion parallel,} or {\em BTP} for brevity, which means that the Bismut connection has parallel torsion tensor. For $n\geq 3$, BTP metrics can be balanced (and non-K\"ahler). In this paper, we give a detailed description to characterize all compact, balanced BTP threefolds.
In this article, we study Hermitian manifolds whose Bismut connection has parallel torsion, which will be called {\em Bismut torsion parallel manifolds,} or {\em BTP} manifolds for brevity. We obtain a necessary and sufficient condition characterizing BTP manifolds in terms of Bismut curvature tensor alone (Theorem 1.1). We also present examples and discuss some general properties for BTP manifolds, as well as give a classification result for non-balanced BTP threefolds (Theorem 1.16).
This paper is a sequel to our studies \cite{ZZ} and \cite{YZZ} on Bismut K\"ahler-like manifolds, or {\em BKL} manifolds for short. We will study the structural theorems for {\em BKL} manifolds, prove a conjecture raised in \cite{YZZ} which states that any {\em BKL} manifold that is Bismut Ricci flat must be Bismut flat, and give complete classifications of {\em BKL} manifolds in dimension $4$ and $5$.
In this article, we study Hermitian manifolds whose Bismut-Strominger connection has parallel torsion tensor, which will be called {\em Bismut torsion parallel manifolds,} or {\em BTP} manifolds for short. We obtain a necessary and sufficient condition characterizing this class in terms of the curvature tensor. In particular, Bismut flat or Bismut K\"ahler-like manifolds are {\em BTP} manifolds, known by our earlier results. We also obtain structural results for non-balanced {\em BTP} manifolds, and classification theorems for non-balanced {\em BTP} threefolds and balanced {\em BTP} threefolds.
In a paper by Angella, Otal, Ugarte, and Villacampa, the authors conjectured that on a compact Hermitian manifold, if a Gauduchon connection other than Chern or Strominger is K\"ahler-like, then the Hermitian metric must be K\"ahler. They also conjectured that if two Gauduchon connections are both K\"ahler-like, then the metric must be K\"ahler. In this paper, we discuss some partial answers to the first conjecture, and give a proof to the second conjecture. In the process, we discovered an interesting `duality' phenomenon amongst Gauduchon connections, which seems to be intimately tied to the question, though we do not know if there is any underlying reason for that from physics.
Let the pair $(\mathfrak{g},J)$ be a nilpotent Lie algebra $\mathfrak{g}$ (NLA for short) endowed with a nilpotent complex structure $J$. In this paper, motivated by a question in the work of Cordero, Fern\'andez, Gray and Ugarte, we prove that $2\leq \nu(J) \leq 3$ for $(\mathfrak{g},J)$ when $\nu(\mathfrak{g})=2$, where $\nu(\mathfrak{g})$ is the step of $\mathfrak{g}$ and $\nu(J)$ is the unique smallest integer such that $\mathfrak{a}(J)_{\nu(J)}=\mathfrak{g}$ as in Definition 1 and 8 of the paper by Cordero, Fern\'andez, Gray and Ugarte. When $\nu(\mathfrak{g})=3$, for arbitrary $n \geq 3$, there exists a pair $(\mathfrak{g},J)$ such that $\nu(J)=\dim_{\mathbb{C}}\mathfrak{g}=n$, for which we call the $J$ in the pair $(\mathfrak{g},J)$, satisfying $\nu(J)=\dim_{\mathbb{C}}\mathfrak{g}=n$, a maximal nilpotent (MaxN for short) complex structure. The algebraic dimension of a nilmanifold endowed with a left invariant MaxN complex structure is discussed. Furthermore, a structure theorem is proved for the pair $(\mathfrak{g},J)$, where $\nu(\mathfrak{g})=3$ and $J$ is a MaxN complex structure.
We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the second author. As a direct corollary of the extension formulas, we prove several deformation invariance theorems for Hodge numbers on some certain classes of complex manifolds, without use of Fr\"{o}licher inequality or the topological invariance of Betti numbers.
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 $(M^n,g)$ is always pluriclosed, and when the manifold is compact and $g$ is not K\"ahler, 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-K\"ahler 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, given 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 $n\geq 3$, and the manifold does not admit any Hermitian symplectic metric.
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.
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. What we actually showed is a bit more: for any given Hermitian manifold, the Strominger K\"ahler-like condition is equivalent to the pluriclosedness of the metric plus the parallelness of the torsion.
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 $(p,p+1)$-th mild $\partial\bar\partial$-lemma under small differentiable deformations.
By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira-Spencer's local stability theorem of K\"ahler structures. We also obtain two new local stability theorems, one of balanced structures on an $n$-dimensional balanced manifold with the $(n-1,n)$-th mild $\partial\bar\partial$-lemma by power series method and the other one on $p$-K\"ahler structures with the deformation invariance of $(p,p)$-Bott-Chern numbers.
We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the first author. As direct corollaries, we prove several deformation invariance theorems for Hodge numbers. Moreover, we also study the Gauduchon cone and its relation with the balanced cone in the K\"ahler case, and show that the limit of the Gauduchon cone in the sense of D. Popovici for a generic fiber in a K\"ahlerian family is contained in the closure of the Gauduchon cone for this fiber.
In this paper, by using the Kuranishi coordinates on the Teichmüller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential geometric proofs of the Torelli theorems, both local and global, for Riemann surfaces.