Plurisigned hermitian metrics

Let $(X,\omega)$ be a compact hermitian manifold of dimension $n$. We study the asymptotic behavior of Monge-Amp\`ere volumes $\int_X (\omega+dd^c \f)^n$, when $\omega+dd^c \f$ varies in the set of hermitian forms that are $dd^c$-cohomologous to $\omega$. We show that these Monge-Amp\`ere volumes are uniformly bounded if $\omega$ is "strongly pluripositive", and that they are uniformly positive if $\omega$ is "strongly plurinegative". This motivates the study of the existence of such plurisigned hermitian metrics. We analyze several classes of examples (complex parallelisable manifolds, twistor spaces, Vaisman manifolds) admitting such metrics, showing that they cannot coexist. We take a close look at $6$-dimensional nilmanifolds which admit a left-invariant complex structure, showing that each of them admit a plurisigned hermitian metric, while only few of them admit a pluriclosed metric. We also study $6$-dimensional solvmanifolds with trivial canonical bundle.