Volumes of Bott–Chern Classes

Abstract 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 Fujiki class $$\mathcal {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> if and only if it has the bounded mass property —i.e., its Monge–Ampère volumes are bounded above—and there exists a closed Bott–Chern class with positive volume. This yields a positive answer to a conjecture of Boucksom–Demailly–Păun. 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 basic results of Boucksom–Eyssidieux–Guedj–Zeriahi.