Special non-K\"ahler metrics -- old and new

TL;DR

This paper reviews various non-K"ahler metrics on compact complex manifolds, discussing their coexistence, deformation behavior, and construction methods, highlighting recent developments and classical results.

Contribution

It provides a comprehensive account of old and new results on non-K"ahler metrics, including construction techniques and deformation analysis.

Findings

Existence of diverse non-K"ahler metrics on compact complex manifolds.

Behavior of these metrics under deformations and blow-ups.

Construction methods for nilmanifolds with specific metric properties.

Abstract

We give an account of old and new results concerning many types of non-K\"ahler metrics, with focus on the problem of their coexistence on compact complex manifolds, and their behaviour at deformations and blow-up. We also describe a mechanism that several authors have used to construct examples of nilmanifolds admitting metrics with certain properties.