Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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D. Angella - V. Guedj - C. H. Lu

Plurisigned hermitian metrics

created by daniele on 12 Jul 2022
modified on 21 Jul 2023


Accepted Paper

Inserted: 12 jul 2022
Last Updated: 21 jul 2023

Journal: Trans. Am. Math. Soc.
Volume: 376
Pages: 4631-4659
Year: 2023
Doi: 10.1090/tran/8916

ArXiv: 2207.04705 PDF


Let $(X,\omega)$ be a compact hermitian manifold of dimension $n$. We study the asymptotic behavior of Monge-Amp\`ere volumes $\int_X (\omega+dd^c \varphi)^n$, when $\omega+dd^c \varphi$ 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.

Tags: PRIN2017-MFDS

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