Constant holomorphic sectional curvature conjecture and Fino-Vezzoni conjecture

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.