arXiv subjects
Publications and source records attributed to Fangyang Zheng.
We classify compact Hermitian manifolds with flat Levi-Civita connection. This is equivalent to the classification of all orthogonal complex structures on flat tori. It generalized the work of Khan, Yang, and Zheng in 2017 where they solved the case in complex dimension three.
Let $M$ be a closed K\"ahler surface. We prove that every Riemannian metric $g$ on $M$ with $\operatorname{Sc}_g\geq-\lambda^2$, where $\lambda\geq 0$, satisfies $$ \lVert M\rVert\leq \frac{27}{2}\,\lambda^4\operatorname{vol}_g(M). $$ This proves Gromov's quantitative scalar-curvature--simplicial-volume conjecture for closed K\"ahler surfaces. We also construct infinitely many non-K\"ahler symplectic 4-manifolds of general type with positive simplicial volume for which the same estimate holds.
We study the Fino-Vezzoni conjecture on compact complex manifolds carrying a balanced Bismut torsion-parallel Hermitian metric. We prove that if such a manifold also admits a pluriclosed Hermitian metric, then it admits a K\"ahler metric.
AAn old conjecture in non-K\"ahler geometry states that any compact Hermitian manifold with constant Chern holomorphic sectional curvature must be either K\"ahler or Chern flat. The conjecture is known to be true in dimension 2 but still open in dimensions 3 or higher, except for several special classes of Hermitian manifolds. For the important class of locally conformally K\"ahler manifolds, the conjecture was proved by H. Chen, L. Chen, and Nie in 2021 when the constant holomorphic sectional curvature is non-positive and the remaining case was solved recently by Huang and Wan using the result of Kamishima on Bochner-K\"ahler manifolds. In this article, we use their technique to answer similar questions for locally conformally K\"ahler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature.
An old conjecture in non-K\"ahler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be K\"ahler (when the constant is non-zero) or Chern flat (when the constant is zero). It is known to be true in complex dimension $2$ by the work of Balas and Gauduchon in 1985 (when the constant is negative or zero) and Apostolov, Davidov and Muskarov in 1996 (in general). In dimension $3$ or higher, the conjecture is only known in some special cases, such as for all twistor spaces by the work of Davidov, Grantcharov, and Muskarov, or for the locally conformally K\"ahler case (when the constant is negative or zero) by the work of H. Chen, L. Chen and Nie, or for all non-balanced Bismut torsion parallel (BTP) manifolds by the work of S. Chen and Zheng, where they also showed that the conjecture holds for all balanced BTP threefolds, utilizing a classification result by Zhao and Zheng. In this article, we show that the conjecture is valid for all balanced BTP manifolds in complex dimension $4$. The interesting part is that while balanced BTP manifolds are highly restrictive, the classification is still lacking in dimensions $4$ or higher, and this study might shed some light on the structure of balanced BTP manifolds in dimension $4$.
In this short essay, we will survey on two conjectures in non-Kähler geometry: the constant holomorphic sectional curvature conjecture and the Fino-Vezzoni conjecture. We aim at the broad audience and assume no expertise in non-Kähler geometry. We will discuss the history and recent developments on these two typical conjectures in the field.
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-Kähler). In this paper, we give a detailed description to characterize all compact, balanced BTP threefolds.
An old conjecture in non-Kähler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be Kähler (when the constant is non-zero) or Chern flat (when the constant is zero). It is known to be true in complex dimension $2$ by the work of Balas and Gauduchon in 1985 (when the constant is negative or zero) and Apostolov, Davidov and Muskarov in 1996 (when the constant is positive). In dimension $3$ or higher, the conjecture is only known in some special cases, such as the locally conformally Kähler case (when the constant is negative or zero) by the work of Chen, Chen and Nie, or for complex nilmanifolds with nilpotent $J$ by the work of Li and the second named author. In this note, we confirm the above conjecture for all non-balanced Bismut torsion parallel (BTP) manifolds. Here the BTP condition means that the Bismut connection has parallel torsion. In particular, the conjecture is valid for all Vaisman manifolds.
In a recent work, Kai Tang conjectured that any compact Hermitian manifold with non-zero constant mixed curvature must be Kähler. He confirmed the conjecture in complex dimension $2$ and for Chern Kähler-like manifolds in general dimensions. In this paper, we verify his conjecture for several special types of Hermitian manifolds, including complex nilmanifolds, solvmanifolds with complex commutators, almost abelian Lie groups, and Lie algebras containing a $J$-invariant abelian ideal of codimension $2$. We also verify the conjecture for all compact balanced threefolds when the Bismut connection has parallel torsion. These results provide partial evidence towards the validity of Tang's conjecture.
In 1976, Milnor classified all Lie groups admitting a flat left-invariant metric. They form a special type of unimodular 2-step solvable groups. Considering Lie groups with Hermitian structure, namely, a left-invariant complex structure and a compatible left-invariant metric, in 2006, Barberis-Dotti-Fino obtained among other things full classification of all Lie groups with Hermitian structure that are K\"ahler and flat. In this note, we examine Lie groups with a Hermitian structure that are flat, and show that they actually must be K\"ahler, or equivalently speaking, a flat Hermitian Lie algebra is always K\"ahler. In the proofs we utilized analysis on the Hermitian geometry of 2-step solvable Lie groups developed by Freibert-Swann and by Chen and the second named author.
We consider the conjecture of Chen and Nie concerning the space forms for canonical metric connections of compact Hermitian manifolds. We verify the conjecture for two special types of Hermitian manifolds: complex nilmanifolds with nilpotent $J$, and non-balanced Bismut torsion-parallel manifolds.
A long-standing conjecture in non-K\"ahler geometry states that if the Chern (or Levi-Civita) holomorphic sectional curvature of a compact Hermitian manifold is a constant $c$, then the metric must be K\"ahler when $c\neq 0$ and must be Chern (or Levi-Civita) flat when $c=0$. The conjecture is known to be true in dimension 2 by the work of Balas-Gauduchon, Sato-Sekigawa, and Apostolov-Davidov-Muskarov in the 1980s and 1990s. In dimension 3 or higher, the conjecture is still open except in some special cases, such as for all twistor spaces by Davidov-Grantcharov-Muskarov, for locally conformally K\"ahler manifolds (when $c\leq 0$) by Chen-Chen-Nie, etc. In this short note, we consider compact quotients $G/\Gamma$ where $G$ is a Lie group equipped with a left-invariant complex structure and a compatible left-invariant metric, and $\Gamma$ is a discrete subgroup. We confirm the conjecture when the Lie algebra ${\mathfrak g}$ of $G$ either is almost abelian, or contains a $J$-invariant abelian ideal of codimension 2.
An old open question in non-Kähler geometry predicts that any compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler or Chern flat. The conjecture is known to be true in dimension $2$ due to the work by Balas-Gauduchon and Apostolov-Davidov-Muskarov in the 1980s and 1990s, but is still open in dimensions $3$ or higher, except in several special cases. The difficulty in this quest for `Hermitian space forms' is largely due to the algebraic complicity or lack of symmetry for the curvature tensor of a general Hermitian metric. In this article, we confirm the conjecture for all solvmanifolds with complex commutator, extending earlier result on nilmanifolds by Li and the second named author.
In our previous work, we introduced a special type of Hermitian metrics called {\em torsion-critical,} which are non-K\"ahler critical points of the $L^2$-norm of Chern torsion over the space of all Hermitian metrics with unit volume on a compact complex manifold. In this short note, we restrict our attention to the class of compact Chern flat manifolds, which are compact quotients of complex Lie groups equipped with compatible left-invariant metrics. Our main result states that, if a Chern flat metric is torsion-critical, then the complex Lie group must be semi-simple, and conversely, any semi-simple complex Lie group admits a compatible left-invariant metric that is torsion-critical.
A Hermitian-symplectic metric is a Hermitian metric whose Kähler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be Kählerian (i.e., admitting a Kähler metric). The conjecture is known to be true in dimension $2$ but is still open in dimensions $3$ or higher. In this article, we confirm the conjecture for all 2-step solvmanifolds, namely, compact quotients of 2-step solvable Lie groups by discrete subgroups. In the proofs, we adopted a method of using special {\em non-unitary} frames, which enabled us to squeeze out some hidden symmetries to make the proof go through. Hopefully the technique could be further applied.
A Hermitian-symplectic metric is a Hermitian metric whose Kähler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be Kählerian (i.e., admitting a Kähler metric). The conjecture is known to be true in dimension $2$ but is open in dimensions $3$ or higher in general, except in a number of special situations, such as twistor spaces (Verbitsky), Fujiki ${\mathcal C}$ spaces (Chiose), Vaisman manifolds (Angella-Otiman), etc. For Lie-complex manifolds (namely, compact quotients $G/Γ$ of Lie groups by discrete subgroups with left-invariant complex structures), the conjecture has also been confirmed in a number of special cases, including when $G$ is nilpotent (Enrietti-Fino-Vezzoni), when $G$ is completely solvable (Fino-Kasuya), or when $J$ is abelian (Fino-Kasuya-Vezzoni), or $G$ is almost abelian (Fino-Kasuya-Vezzoni, Fino-Paradiso), etc. In this article, we conduct a detailed case analysis and confirm Streets-Tian Conjecture for $G$ whose Lie algebra contains an abelian ideal of codimension $2$. Such Lie algebras are always solvable of step at most $3$, but are not $2$-step solvable and not completely solvable in general. Our approach is explicit in nature by describing both the Hermitian-symplectic metrics on such Lie algebras and the pathways of deforming them into Kähler ones, in hope of advancing our understanding of the subtlety and intricacy of this interesting conjecture in non-Kähler geometry.
A Hermitian-symplectic metric is a Hermitian metric whose K\"ahler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be K\"ahlerian (i.e., admitting a K\"ahler metric). The conjecture is known to be true in complex dimension $2$ but is still open in complex dimensions $3$ or higher. In this article, we confirm the conjecture for some special types of compact Hermitian manifolds, including Chern K\"ahler-like manifolds, non-balanced Bismut torsion parallel (BTP) manifolds, and compact quotients of Lie groups whose Lie algebra contains a $J$-invariant abelian ideal of codimension $2$. The last type is a natural generalization to (compact quotients of) almost abelian Lie groups. The non-balanced BTP case contains all Vaisman manifolds and all Bismut K\"ahler-like manifolds as subsets. These results extend some of the earlier works on the topic by Fino, Kasuya, Vezzoni, Angella, Otiman, Paradiso, and others. Our approach is elementary in nature, by giving explicit descriptions of Hermitian-symplectic metrics on such spaces as well as the pathways of deforming them into K\"ahler ones, aimed at illustrating the algebraic complexity and subtlety of Streets-Tian Conjecture.
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).