Volumes of Bott-Chern classes

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.