Title: Volumes of Bott-Chern classes Show Chinese title

Title: Bott-Chern类的体积

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 class $\mathcal{C}$ of Fujiki if and only if it has the $\textit{bounded mass property}$ -- i.e., its Monge-Amp\`ere 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-P\u{a}un-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-Amp\`ere masses, and moreover show the existence of solutions to degenerate complex Monge-Amp\`ere equations in big classes, together with uniform a priori estimates. This extends to the hermitian context fundamental results of Boucksom-Eyssidieux-Guedj-Zeriahi. Show Chinese abstract

Abstract: 我们研究任意紧复流形$X$上超越的可能非闭的 Bott-Chern$(1,1)$类的体积。我们证明后者属于 Fujiki 的类$\mathcal{C}$当且仅当它具有$\textit{bounded mass property}$—— 即其 Monge-Ampère 体积有统一的上界 —— 并且存在一个具有正体积的闭 Bott-Chern 类。这为 Demailly-Păun-Boucksom 的一个猜想给出了肯定的回答。 为此，我们将电流的非纯量积的概念扩展到厄米特情形，允许后者仅为${\it quasi}$-${\it closed}$和${\it quasi}$-${\it positive}$。 我们建立了蒙日-安培质量的准单调性性质，并且还证明了在大类中退化复蒙日-安培方程解的存在性，以及一致的先验估计。 这将布克斯姆-埃西迪埃-古德齐-泽里亚希的基本结果扩展到了厄米特情形。