Volumes of Bott-Chern classes
arxiv(2024)
摘要
We study the volumes of transcendental and possibly non-closed Bott-Chern
(1,1)-classes on an arbitrary compact complex manifold X. We show that the
latter belongs to the class 𝒞 of Fujiki if and only if it has the
bounded mass property – i.e., its Monge-Ampère volumes have a
uniform upper-bound – and there exists a closed Bott-Chern class with positive
volume. This yields a positive answer to a conjecture of
Demailly-Păun-Boucksom. To this end we extend to the hermitian context the
notion of non-pluripolar products of currents, allowing for the latter to be
merely quasi-closed and quasi-positive. We
establish a quasi-monotonicity property of Monge-Ampère masses, and moreover
show the existence of solutions to degenerate complex Monge-Ampère equations
in big classes, together with uniform a priori estimates. This extends to the
hermitian context fundamental results of Boucksom-Eyssidieux-Guedj-Zeriahi.
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