Boucksom–Demailly–Păun–Peternell conjecture on transcendental holomorphic Morse inequalities

Let $X$ be an $n$-dimensional compact complex manifold. For a real $d$-closed $(1,1)$-form $\alpha$, let $X(\alpha,\leq 1)$ be the set where $\alpha$ has at most one negative eigenvalue, and write $\{\alpha\}$ for its Bott–Chern class. Let $\operatorname{vol}(\{\alpha\})$ denote the volume of this class. Also, let $\{\beta\}$ be a nef cohomology class of type $(1,1)$ on $X$. Boucksom–Demailly–Păun–Peternell conjecture. (i) If

$$\int_{X(\alpha,\leq 1)}\alpha^n>0,$$

then $\{\alpha\}$ contains a Kähler current and

$$\operatorname{vol}(\{\alpha\})\geq\int_{X(\alpha,\leq 1)}\alpha^n.$$

(ii) If $\{\alpha\}$ and $\{\beta\}$ satisfy

$$\{\alpha\}^n-n\{\alpha\}^{n-1}\cdot\{\beta\}>0,$$

then the Bott–Chern class $\{\alpha-\beta\}$ contains a Kähler current and

$$\operatorname{vol}(\{\alpha-\beta\})\geq\{\alpha\}^n-n\{\alpha\}^{n-1}\cdot\{\beta\}.$$

These conjectures formulate transcendental holomorphic Morse inequalities for compact complex manifolds and relate positivity of $(1,1)$-classes to the existence of Kähler currents. The paper proves only a weak version of Demailly's conjecture, so the stated Boucksom–Demailly–Păun–Peternell conjecture is not resolved here.