Search arXiv⌕ Search

arXiv subjects

Fangyang Zheng

Publications and source records attributed to Fangyang Zheng.

At least 19 recordsLinked to original sources
arXiv2026

Balanced Bismut torsion-parallel manifold with constant holomorphic sectional curvature

A long-lasting conjecture in non-Kähler geometry says that any compact Hermitian manifold with constant holomorphic sectional curvature must be either Kähler or Chern flat. The conjecture is known to be true in dimension 2 but remains open in dimensions 3 or higher, except in several special cases. In this article we consider compact Hermitian manifolds that are Bismut torsion-parallel (BTP for brevity), which means that the Bismut connection has parallel torsion. For BTP manifolds that are not balanced, the conjecture was proved by Chen and Zheng. They also confirmed the conjecture for balanced BTP manifolds in dimension 3, utilizing a classification result for such threefolds by Zhao and Zheng. In a recent work, we confirmed the conjecture for balanced BTP manifolds in dimension 4 through a detailed case-by-case analysis. In this article, we establish the conjecture for balanced BTP manifolds in all dimensions. We also disprove the full-rank $B$ conjecture by constructing a compact balanced BTP fivefold with full-rank $B$ that is neither Kähler nor Chern flat; its Chern holomorphic sectional curvature is nonconstant.

math.DG↗
arXiv2026

On Compact Hermitian Surfaces With Positive Strominger-Bismut Sectional Curvature

In an earlier work, Yau and Zheng proposed a Hermitian analogue of a weak form of the Frankel conjecture, which states that any compact Hermitian manifold with positive Strominger-Bismut sectional curvature must be biholomorphic to the complex projective space. In this article, we confirm the conjecture in complex dimension two. This is achieved by a vanishing theorem for anti-self-dual harmonic 2-forms on such surfaces, utilizing a curvature decomposition formula by Ferreira and a refined Kato inequality.

math.DG↗
arXiv2026

Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature

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 (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 Kähler 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 $4$.

math.DG↗
arXiv2026

Orthogonal complex structures on flat tori

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.

math.DG↗
arXiv2026

Simplicial Volume and Scalar Curvature on Closed Kähler Surfaces

Let $M$ be a closed Kähler surface. We prove that every Riemannian metric $g$ on $M$ with $\operatorname{Sc}_g\geq-λ^2$, where $λ\geq 0$, satisfies $$ \lVert M\rVert\leq \frac{27}{2}\,λ^4\operatorname{vol}_g(M). $$ This proves Gromov's quantitative scalar-curvature--simplicial-volume conjecture for closed Kähler surfaces. We also construct infinitely many non-Kähler symplectic 4-manifolds of general type with positive simplicial volume for which the same estimate holds.

math.DG↗
arXiv2026

Hermitian Connections with Parallel Torsion and the Fino--Vezzoni Conjecture

We prove that the Fino--Vezzoni conjecture holds on every compact complex manifold carrying a Hermitian connection with parallel torsion. More precisely, if a compact connected complex manifold carries a Hermitian metric $g$ and a Hermitian connection $D$ with $DT^D=0$, then the coexistence of a balanced metric and a pluriclosed metric forces the existence of a $D$-parallel Kähler metric. The proof combines a holonomy symmetrization onto $D$-parallel forms with a finite-dimensional volume-maximization argument. In particular, the conjecture holds on every compact Bismut torsion-parallel manifold. In the non-balanced case, the obstruction of Zhao--Zheng yields the stronger conclusion that balanced and pluriclosed metrics cannot coexist.

math.DG↗
arXiv2026

Curvature property on Hermitian Lie algebras with abelian ideals of codimension two

Let $(\mathfrak g,J,g)$ be a unimodular Hermitian Lie algebra containing an abelian ideal $\mathfrak a$ of real codimension two. We study the curvature behaviour of $g$ and show that, if $g$ has constant Chern holomorphic sectional curvature, then it must be Chern flat. We also show that, for every canonical metric connection $D_s^r$ of $g$ other than the Chern connection, if $D_s^r$ has constant holomorphic sectional curvature, then $g$ is Kähler flat, and in this case $\mathfrak g/\mathfrak a$ is abelian.

math.DG↗
arXiv2026

Locally conformally Kähler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature

AAn old conjecture in non-Kähler geometry states that any compact Hermitian manifold with constant Chern holomorphic sectional curvature must be either Kähler 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 Kähler 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-Kähler manifolds. In this article, we use their technique to answer similar questions for locally conformally Kähler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature.

math.DG↗
arXiv2026

Curvature characterization of Hermitian manifolds with Bismut parallel torsion

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).

math.DG↗
arXiv2025

Constant holomorphic sectional curvature conjecture and Fino-Vezzoni conjecture

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.

math.DG↗
arXiv2025

On balanced Hermitian threefolds with parallel Bismut torsion

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.

math.DG↗
arXiv2025

Bismut torsion parallel metrics with constant holomorphic sectional curvature

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.

math.DG↗
arXiv2025

On Hermitian manifolds with constant mixed curvature

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.

math.DG↗
arXiv2025

Flat Hermitian Lie algebras are Kähler

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 Kähler and flat. In this note, we examine Lie groups with a Hermitian structure that are flat, and show that they actually must be Kähler, or equivalently speaking, a flat Hermitian Lie algebra is always Kähler. 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.

math.DG↗
arXiv2025

Streets-Tian Conjecture on several special types of Hermitian manifolds

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 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 Kähler-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 Kähler-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 Kähler ones, aimed at illustrating the algebraic complexity and subtlety of Streets-Tian Conjecture.

math.DG↗
arXiv2025

Canonical metric connections with constant holomorphic sectional curvature

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.

math.DG↗
arXiv2025

A note on almost abelian groups with constant holomorphic sectional curvature

A long-standing conjecture in non-Kähler 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 Kähler 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 Kähler manifolds (when $c\leq 0$) by Chen-Chen-Nie, etc. In this short note, we consider compact quotients $G/Γ$ where $G$ is a Lie group equipped with a left-invariant complex structure and a compatible left-invariant metric, and $Γ$ 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.

math.DG↗