Divisorial Zariski decomposition and algebraic Morse inequalities

In this note we use the divisorial Zariski decomposition to give a more intrinsic version of the algebraic Morse inequalities.Comment: In this version we correct some misprints

Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle

We show that if a compact complex manifold admits a K\"ahler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.Comment: 12 pages, no figures, final version, to appear on J. Differential Geo

A remark on the codimension of the Green-Griffiths locus of generic projective hypersurfaces of high degree

We show that for every smooth generic projective hypersurface $X\subset\mathbb P^{n+1}$, there exists a proper subvariety $Y\subsetneq X$ such that $\operatorname{codim}_X Y\ge 2$ and for every non constant holomorphic entire map $f\colon\mathbb C\to X$ one has $f(\mathbb C)\subset Y$, provided $\deg X\ge 2^{n^5}$. In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in $\mathbb P^4$.Comment: 7 pages, no figures, comments are welcome. Corrected typos, added a small section with an open question, references updated. Final version, to appear on J. Reine Angew. Math

Monge-Ampère measures on contact sets

Let $(X, \omega)$ be a compact K\"ahler manifold of complex dimension n and $\theta$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class $\{ \theta \}\in H^{1,1}(X, \mathbb{R})$ is pseudoeffective. Let $\varphi$ be a $\theta$-psh function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $\omega$ such that $\varphi \leq f.$ Then the non-pluripolar measure $\theta_\varphi^n:= (\theta + dd^c \varphi)^n$ satisfies the equality: $${\bf{1}}_{\{ \varphi = f \}} \ \theta_\varphi^n = {\bf{1}}_{\{ \varphi = f \}} \ \theta_f^n,$$ where, for a subset $T\subseteq X$, ${\bf{1}}_T$ is the characteristic function. In particular we prove that \[ \theta_{P_{\theta}(f)}^n= { \bf {1}}_{\{P_{\theta}(f) = f\}} \ \theta_f^n\qquad {\rm and }\qquad \theta_{P_\theta[\varphi](f)}^n = { \bf {1}}_{\{P_\theta[\varphi](f) = f \}} \ \theta_f^n. \