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

The goal of this work is to pursue the study pseudo-effective line bundles and vector bundles. Our first result a generalization Hard Lefschetz theorem for cohomology with values in bundle. map shown be surjective when (and, general, only when) bundle twisted by its multiplier ideal sheaf. This has several geometric applications, e.g. compact Kähler manifolds canonical or anti-canonical Another concern understand pseudo-effectivity more algebraic terms. In direction, we introduce concept an...

We first prove a strengthening of Miyaoka's generic semi-positivity theorem: the quotients tensor powers cotangent bundle non-uniruled complex projective manifold X have pseudo-effective (instead generically nef) determinant. A consequence is that general type if its contains subsheaf with 'big' Among other applications, we deduce universal cover not covered by compact positive-dimensional analytic subsets, then χ(O )≠0. finally show L numerically trivial line on X, and K +L ℚ-effective, so...

Given a quasi-projective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite \'etale covers from smooth locus $X_{\mathrm{reg}}$ of $X$ itself. A simplified version our main results states that there exists Galois cover $Y \rightarrow X$, ramified over singularities $X$, such fundamental groups $Y$ and $Y_{\mathrm{reg}}$ agree. In particular, every extends an $Y$. As first major application, show flat holomorphic bundle defined on is all...

This paper extends a number of known results on slope-semistable sheaves from the classical case to setting where polarizations are given by movable curve classes. As applications, we obtain new flatness for reflexive singular varieties, as well characterization finite quotients Abelian varieties via Chern class condition.

Abstract In a follow-up to our paper [Threefolds with big and nef anticanonical bundles I, Math. Ann., 2005, 333(3), 569–631], we classify smooth complex projective threefolds Xwith −K X but not ample, Picard number γ(X) = 2, whose map is small. We assume also that the Mori contraction of its flop + are both birational.

Let X be a compact Kähler threefold with terminal singularities such that K is nef.We prove semiample.

<!-- *** Custom HTML --> The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover decomposes into a product of torus, and irreducible, simply-connected Calabi-Yau– holomorphic-symplectic manifolds. decomposition the part corresponds to tangent direct sum whose summands are integrable stable respect polarisation. Building on recent extension theorems for differential forms singular spaces,...

The goal of this paper is to make a surprising connection between several central conjectures in algebraic geometry: the Nonvanishing Conjecture, Abundance and Semiampleness Conjecture for nef line bundles on K-trivial varieties.

We prove a Kawamata-Viehweg vanishing theorem on normal compact Kähler space X: if L is nef line bundle with L2 ≠ 0, then H>q(X,KX+L) = 0 for q ≥ dim X − 1. As an application we complete part of the abundance minimal threefolds: threefold, Kodaira dimension κ(X) nonnegative.

We establish the Miyaoka-Yau inequality in terms of orbifold Chern classes for tangent sheaf any complex projective variety general type with klt singularities and nef canonical divisor.In case equality is a ained at worst terminal singularities, we prove that associated model quotient unit ball by discrete group action.C 1. Introduction 1 2. Notation standard facts 5 Part I. Foundational material 9 3. Q-varieties Q-Chern 4. Sheaves operators 16 5.Higgs sheaves 19 II.Miyaoka-Yau Inequality...

We prove that if (X,\Delta) is a threefold pair with mild singularities such -(K_{X}+\Delta) nef, then the numerical class of effective.

The aim of this paper is to describe the structure Fano bundles in dimension > 4. Introduction.In rank 2 vector E on projective spaces Ψ n and quadrics Q are investigated which enjoy additional property that their projectized Ψ(E) manifolds, i.e. have negative canonical bundles.Such shortly called bundles.Up 3 completely classified by [SW], [SW"], [SSW].The Namely we prove following MAIN THEOREM.Let be a bundle or , Then up some explicit exceptions 4 Q$ (see ex.(2.1), (2.2), (2.3)), splits...