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Constant holomorphic sectional curvature conjecture and Fino-Vezzoni conjecture

Fangyang Zheng

TL;DR

The paper surveys two central problems in non-Kähler geometry: whether a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler (or flat when $c=0$) and whether a compact non-Kähler manifold can simultaneously admit balanced and pluriclosed metrics. It canvasses a broad set of techniques and settings, from twistor spaces and lcK geometry to Lie-Hermitian and nilmanifold cases, compiling known results and identifying remaining open cases. The work highlights dimension- and structure-specific progress (complete resolution in complex dimension $2$, partial results for lcK, Chern/Riemannian/Kähler-like connections, and BTP scenarios) and recent advances toward the Fino–Vezzoni conjecture, including new classifications and rigidity phenomena in Lie-Hermitian settings. Overall, the survey clarifies where the conjectures hold, what structural obstructions arise in non-Kähler geometry, and how recent methods narrow the gap toward full resolution with implications for complex and differential geometry.

Abstract

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.

Constant holomorphic sectional curvature conjecture and Fino-Vezzoni conjecture

TL;DR

The paper surveys two central problems in non-Kähler geometry: whether a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler (or flat when ) and whether a compact non-Kähler manifold can simultaneously admit balanced and pluriclosed metrics. It canvasses a broad set of techniques and settings, from twistor spaces and lcK geometry to Lie-Hermitian and nilmanifold cases, compiling known results and identifying remaining open cases. The work highlights dimension- and structure-specific progress (complete resolution in complex dimension , partial results for lcK, Chern/Riemannian/Kähler-like connections, and BTP scenarios) and recent advances toward the Fino–Vezzoni conjecture, including new classifications and rigidity phenomena in Lie-Hermitian settings. Overall, the survey clarifies where the conjectures hold, what structural obstructions arise in non-Kähler geometry, and how recent methods narrow the gap toward full resolution with implications for complex and differential geometry.

Abstract

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.

Paper Structure

This paper contains 7 sections, 31 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Examples of non-Kähler manifolds
  3. Almost complex manifolds
  4. Compact Kähler manifolds
  5. Non-Kähler manifolds
  6. Constant holomorphic sectional curvature conjecture
  7. Fino-Vezzoni conjecture

Key Result

Theorem 1

A closed orientable $4$-manifold $N^4$ admits an almost complex structure $J$ if and only if there exists a class $h\in H^2(N, {\mathbb Z})$ such that $h^2=3\sigma +2\chi$ and $h\equiv w_2$ mod $2$, where $\sigma$ is the signature, $\chi$ is the Euler number, and $w_2$ is the second Stiefel-Whitney

Theorems & Definitions (45)

  • Conjecture 1: Constant Holomorphic Sectional Curvature Conjecture
  • Conjecture 2: Fino-Vezzoni Conjecture
  • Theorem 1: Wu
  • Example : 1
  • Theorem 2
  • Conjecture 3: Yau
  • Theorem 3: Newlander-Nirenberg
  • Conjecture 4: Peternell
  • Example : 2
  • Example : 3
  • ...and 35 more