International audienceWe prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of " movable curves " , which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension

The goal of this work is to give a precise numerical description of the Kähler cone of a compact Kähler manifold. Our main result states that the Kähler cone depends only on the intersection form of the cohomology ring, the Hodge structure and the homology classes of analytic cycles: if X is a compact Kähler manifold, the Kähler cone K of X is one of the connected components of the set P of real (1, 1)-cohomology classes {α} which are numerically positive on analytic cycles, i.e. Y α p > 0 for every irreducible analytic set Y in X, p = dim Y . This result is new even in the case of projective manifolds, where it can be seen as a generalization of the well-known Nakai-Moishezon criterion, and it also extends previous results by Campana-Peternell and Eyssidieux. The principal technical step is to show that every nef class {α} which has positive highest self-intersection number X α n > 0 contains a Kähler current; this is done by using the Calabi-Yau theorem and a mass concentration technique for Monge-Ampère equations. The main result admits a number of variants and corollaries, including a description of the cone of numerically effective (1, 1)-classes and their dual cone. Another important consequence is the fact that for an arbitrary deformation X → S of compact Kähler manifolds, the Kähler cone of a very general fibre X t is "independent" of t, i.e. invariant by parallel transport under the (1, 1)-component of the Gauss-Manin connection.

Abstract. The main result of the present article is a (practically optimal) criterion for the pseudoeffectivity of the twisted relative canonical bundles of surjective projective maps. Our theorem has several applications in algebraic geometry; to start with, we obtain the natural analytic generalization of some semipositivity results due to E. Viehweg and F. Campana. As a byproduct, we give a simple and direct proof of a recent result due to C. Hacon-J. McKernan and S. Takayama concerning the extension of twisted pluricanonical forms. More applications will be offered in the sequel of this article. §0 IntroductionIn this article our primary goal is to establish some positivity results concerning the twisted relative canonical bundle of projective morphisms.Let X and Y be non-singular projective manifolds, and let p : X → Y be a surjective projective map, whose relative dimension is equal to n. Consider also a line bundle L over X, endowed with a -possibly singular-metric h = e −φ , such that the curvature current is semipositive. We denote by I(h) the multiplier ideal sheaf of h (see e.g. [10], [21], [25]). Let X y be the fiber of p over a point y ∈ Y , such that y is not a critical value of p. We also assume at first that the restriction of the metric φ to X y is not identically −∞. Under these circumstances, the space of (n, 0) forms L-valued on X y which belong to the multiplier ideal sheaf of the restriction of the metric h is endowed with a natural L 2 -metric as follows(we use the standard abuse of notation in the relation above). Let us consider an orthonormal basis (uRecall that the bundles K X y and K X/Y |X y are isomorphic. Via this identification (which will be detailed in the paragraph 1) the sections above can be used to define 2 Bergman kernels and the pseudoeffectivity of relative canonical bundles a metric on the bundle K X/Y + L restricted to the fiber X y , called the Bergman kernel metric. This definition immediately extends also to fibers such that the metric φ is identically equal to −∞ on the fiber. In this case the Bergman kernel vanishes identically on the fiber, and the Bergman kernel metric is also equal to −∞ there. LetThen we have the next result, which gives a pseudoeffectivity criterion for the bundle0.1 Theorem. Let p : X → Y be a surjective projective map between smooth manifolds, and let (L, h) be a holomorphic line bundle endowed with a metric h such that: Then the relative Bergman kernel metric of the bundleIt has semipositive curvature current and extends across X \ X 0 to a metric with semipositive curvature current on all of X.Several versions of the theorem above were established by the first author in his series of articles on the plurisubharmonic variation of the Bergman kernels (see [1], [2], [3] and also [22] for the first results in this direction). Let us point out the main improvements we have got in the present article. In the first place, we allow the metric h to be singular. Secondly, the map p is not supposed to be a smooth fibration-this will be crucial for t...

Let X be a non-singular compact Kähler manifold, endowed with an effective divisor D = (1 − β k )Y k having simple normal crossing support, and satisfying β k ∈ (0, 1). The natural objects one has to consider in order to explore the differentialgeometric properties of the pair (X, D) are the so-called metrics with conic singularities. In this article, we complete our earlier work [CGP13] concerning the Monge-Ampère equations on (X, D) by establishing Laplacian and C 2,α,β estimates for the solution of this equations regardless to the size of the coefficients 0 < β k < 1. In particular, we obtain a general theorem concerning the existence and regularity of Kähler-Einstein metrics with conic singularities along a normal crossing divisor.