We prove a conjecture of Boucksom-Demailly-Puaun-Peternell, namely that on a projective manifold $X$ the cone of pseudoeffective classes in $H^{1,1}_{\mathbb{R}}(X)$ is dual to the cone of movable classes in $H^{n-1,n-1}_{\mathbb{R}}(X)$ via the Poincar\'e pairing. This is done by establishing a conjectured transcendental Morse inequality for the volume of the difference of two nef classes on a projective manifold. In an appendix by Boucksom it is shown that the Morse inequality also implies that the volume function is differentiable on the big cone, and one also gets a characterization of the prime divisors in the non-K\"ahler locus of a big class via intersection numbers.

中文翻译：

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射影流形上拟有效锥和动锥的对偶性

我们证明了 Boucksom-Demailly-Puaun-Peternell 的猜想，即在射影流形 $X$ 上，$H^{1,1}_{\mathbb{R}}(X)$ 中的伪有效类的锥是对偶的通过 Poincar\'e 配对到 $H^{n-1,n-1}_{\mathbb{R}}(X)$ 中的可移动类锥。这是通过为射影流形上两个 nef 类的差的体积建立一个推测的超越莫尔斯不等式来完成的。在 Boucksom 的附录中，Morse 不等式还表明体积函数在大锥体上是可微的，并且还可以通过交集获得大类的非 K\"ahler 轨迹中的质因数的表征数字。