Weak transcendental holomorphic Morse inequalities on compact K\"ahler manifolds

Transcendental holomorphic Morse inequalities aim at characterizing the positivity of transcendental cohomology classes of type $(1,1)$. In this paper, we prove a weak version of Demailly's conjecture on transcendental Morse inequalities on compact K\"ahler manifolds. And as a consequence, we partially improve a result of Boucksom-Demailly-Paun-Peternell.Comment: v2, typos corrected, to appear in Annales de l'Institut Fourie

Movable intersection and bigness criterion

In this note, we give a Morse-type bigness criterion for the difference of two pseudo-effective $(1,1)$-classes by using movable intersections. As an application, we give a Morse-type bigness criterion for the difference of two movable $(n-1,n-1)$-classes.Comment: v4: some arguments corrected and simplified, references update

Convexity and Zariski decomposition structure

This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre-Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves defined by the second named author, yielding some fundamental positivity results for curve classes. For varieties with special structures, the Zariski decomposition for curve classes admits an interesting geometric interpretation.Comment: 38 pages; revised half of the previous arXiv:1507.04316 [math.AG