Constant holomorphic sectional curvature conjecture and Fino-Vezzoni conjecture

TL;DR

This paper surveys two major conjectures in non-Kähler geometry, discussing their history, recent progress, and significance for broad mathematical audiences without requiring prior expertise.

Contribution

It provides a comprehensive overview of the constant holomorphic sectional curvature and Fino-Vezzoni conjectures, highlighting recent developments and open problems.

Findings

Summarizes historical context of the conjectures

Reviews recent progress and partial results

Identifies key open questions in non-Kähler geometry

Abstract

In this short essay, we will survey on two conjectures in non-K\"ahler geometry: the constant holomorphic sectional curvature conjecture and the Fino-Vezzoni conjecture. We aim at the broad audience and assume no expertise in non-K\"ahler geometry. We will discuss the history and recent developments on these two typical conjectures in the field.