On the existence of balanced and SKT metrics on nilmanifolds (Q2796714)

Let \(M\) be a complex manifold, and denote by \(J\) its almost complex structure. A Riemannian metric \(g\) is called \(J\)-Hermitian if we have \(g(J ., J .) = g(., .)\). Among \(J\)-Hermitian metrics one can define two types which are of special interest: a metric \(g\) is ``strong Kähler with torsion'' (SKT) if its fundamental form \(\omega\) is \(\partial \overline \partial\)-closed, and it is ``balanced'' if \(\omega\) is \(d^*\)-closed. A metric that is both SKT and balanced is always a Kähler metric, and one expects a similar property at the level of manifolds: a compact complex manifold admitting both an SKT metric and a balanced metric is always Kähler. This conjecture is wide open in general. In this paper the authors prove it for nilmanifolds, i.e., quotients \(G \backslash \Gamma\) where \(G\) is a simply connected nilpotent Lie group and \(\Gamma\) a cocompact lattice. More precisely the authors show that if a nilmanifold admits an SKT metric and a balanced metric, then \(G \backslash \Gamma\) is a complex torus.