2024/06/03 by Boucksom, Sébastien, Guedj, Vincent, Lu, Chinh H. · 3 citations
#Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2406.01090
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 \it quasi-\it closed and \it quasi-\it 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.