Real semisimple Lie groups and balanced metrics

TL;DR

This paper proves the existence of invariant balanced Hermitian metrics with zero Chern scalar curvature on certain non-compact real simple Lie groups and their compact quotients, highlighting differences from compact Lie groups.

Contribution

It establishes the existence of balanced metrics with special curvature properties on non-compact real simple Lie groups and their quotients, a novel result in differential geometry.

Findings

Existence of invariant balanced metrics with zero Chern scalar curvature on G and M.

M does not admit any pluriclosed metric, contrasting with compact Lie groups.

Balanced metrics are shown to exist in cases where pluriclosed metrics do not.

Abstract

Given any non-compact real simple Lie group G of inner type and even dimension, we prove the existence of an invariant complex structure J and a Hermitian balanced metric with vanishing Chern scalar curvature on G and on any compact quotient $M=G/\Gamma$, with $\Gamma$ a cocompact lattice. We also prove that (M,J) does not carry any pluriclosed metric, in contrast to the case of even dimensional compact Lie groups, which admit pluriclosed but not balanced metrics.