Restricted volumes and divisorial Zariski decompositions

We give a relation between the existence of a Zariski decomposition and the behavior of the restricted volume of a big divisor on a smooth (complex) projective variety. Moreover, we give an analytic description of the restricted volume in the line of Boucksom's work. It enables us to define the restricted volume of a transcendental class on a compact K\"ahler manifold in natural way. The relation can be extended to a transcendental class.Comment: 24 pages, to appear in Amer. J. Math, v2: completely revise

On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the numerical dimension of images of such fibrations is zero under the assumption of the abundance conjecture. As an application, we show that any compact Kaehler surface with semi-positive holomorphic sectional curvature is rationally connected, or a complex torus, or a ruled surface over an elliptic curve.Comment: 21pages, comments are welcom

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Koll'ar's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.Comment: 18pages, to appear in Adv. Math. 280 (2015), 188--207, v3: completely revised versio