On the existence of balanced and SKT metrics on nilmanifolds

TL;DR

This paper proves that on nilmanifolds, if a complex manifold admits both SKT and balanced metrics, then it must be Kähler, confirming a conjecture in this specific setting.

Contribution

The paper confirms the conjecture that SKT and balanced metrics imply Kählerness on nilmanifolds, a significant step in understanding complex geometric structures.

Findings

Confirmed the conjecture for nilmanifolds

SKT and balanced metrics imply Kähler on nilmanifolds

Provides insights into complex geometry on nilmanifolds

Abstract

On a complex manifold an Hermitian metric which is simultaneously SKT and balanced has to be necessarily K\"ahler. It has been conjectured that if a compact complex manifold (M,J) has an SKT metric and a balanced metric both compatible with J, then (M, J) is necessarily K\"ahler. We show that the conjecture is true for nilmanifolds.