scholarly journals PSEUDO-EFFECTIVE LINE BUNDLES ON COMPACT KÄHLER MANIFOLDS

2001 ◽  
Vol 12 (06) ◽  
pp. 689-741 ◽  
Author(s):  
JEAN-PIERRE DEMAILLY ◽  
THOMAS PETERNELL ◽  
MICHAEL SCHNEIDER
Keyword(s):  
Line Bundle ◽  
Vector Bundles ◽  
Kähler Manifolds ◽  
Line Bundles ◽  
Kahler Manifolds ◽  
Ideal Sheaf ◽  
Multiplier Ideal Sheaf ◽  
Albanese Map ◽  
Multiplier Ideal ◽  
Hard Lefschetz Theorem

The goal of this work is to pursue the study of pseudo-effective line bundles and vector bundles. Our first result is a generalization of the Hard Lefschetz theorem for cohomology with values in a pseudo-effective line bundle. The Lefschetz map is shown to be surjective when (and, in general, only when) the pseudo-effective line bundle is twisted by its multiplier ideal sheaf. This result has several geometric applications, e.g. to the study of compact Kähler manifolds with pseudo-effective canonical or anti-canonical line bundles. Another concern is to understand pseudo-effectivity in more algebraic terms. In this direction, we introduce the concept of an "almost" nef line bundle, and mean by this that the degree of the bundle is nonnegative on sufficiently generic curves. It can be shown that pseudo-effective line bundles are almost nef, and our hope is that the converse also holds true. This can be checked in some cases, e.g. for the canonical bundle of a projective 3-fold. From this, we derive some geometric properties of the Albanese map of compact Kähler 3-folds.

scholarly journals Numerical dimension and a Kawamata–Viehweg–Nadel-type vanishing theorem on compact Kähler manifolds

2014 ◽  
Vol 150 (11) ◽  
pp. 1869-1902 ◽  
Author(s):  
Junyan Cao
Keyword(s):  
Line Bundle ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Line Bundles ◽  
Kahler Manifolds ◽  
Vanishing Theorem ◽  
Kahler Manifold ◽  
Singular Metrics ◽  
Compact Kähler Manifold

AbstractLet $X$ be a compact Kähler manifold and let $(L,{\it\varphi})$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension for pseudo-effective line bundles with singular metrics, and then discuss the properties of this numerical dimension. Finally, we prove a very general Kawamata–Viehweg–Nadel-type vanishing theorem on an arbitrary compact Kähler manifold.

scholarly journals (k, s)-POSITIVITY AND VANISHING THEOREMS FOR COMPACT KÄHLER MANIFOLDS

2011 ◽  
Vol 22 (04) ◽  
pp. 545-576 ◽  
Author(s):  
QILIN YANG
Keyword(s):  
Line Bundle ◽  
Vector Bundles ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Vanishing Theorems ◽  
Positive Vector ◽  
Holomorphic Vector Bundles ◽  
Positive Line Bundle ◽  
Algebraic Manifolds ◽  
Positive Line

We study the (k, s)-positivity for holomorphic vector bundles on compact complex manifolds. (0, s)-positivity is exactly the Demailly s-positivity and a (k, 1)-positive line bundle is just a k-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for (k, s)-positive vector bundles are proved and the vanishing theorems for k-ample vector bundles on projective algebraic manifolds are generalized to k-positive vector bundles on compact Kähler manifolds.

scholarly journals Chern Class Inequalities on Polarized Manifolds and Nef Vector Bundles

2020 ◽  
Author(s):  
Ping Li ◽  
Fangyang Zheng
Keyword(s):  
Chern Class ◽  
Vector Bundles ◽  
Euler Number ◽  
Type Inequality ◽  
Chern Number ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Bisectional Curvature ◽  
Chern Numbers ◽  
Nef Vector Bundles

Abstract This article is concerned with Chern class and Chern number inequalities on polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with $L$ very ample, our 1st main result is a family of sharp Chern class inequalities. Among them the 1st one is a variant of a classical result and the equality case of the 2nd one is a characterization of hypersurfaces. The 2nd main result is a Chern number inequality on it, which includes a reverse Miyaoka–Yau-type inequality. The 3rd main result is that the Chern numbers of a nef vector bundle over a compact Kähler manifold are bounded below by the Euler number. As an application, we classify compact Kähler manifolds with nonnegative bisectional curvature whose Chern numbers are all positive. A conjecture related to the Euler number of compact Kähler manifolds with nonpositive bisectional curvature is proposed, which can be regarded as a complex analogue to the Hopf conjecture.

scholarly journals Vanishing Theorems on Compact Hyper-kähler Manifolds

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Qilin Yang
Keyword(s):  
Line Bundle ◽  
Nonnegative Integer ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Holomorphic Line Bundle ◽  
Vanishing Theorems ◽  
Vanishing Theorem ◽  
Positive Holomorphic Line Bundle ◽  
Special Case

We prove that if B is a k-positive holomorphic line bundle on a compact hyper-kähler manifold M, then HpM,Ωq⊗B=0 for P>n+[k/2] with q a nonnegative integer. In a special case, k=0 and q=0, we recover a vanishing theorem of Verbitsky’s with a little stronger assumption.

Sign in / Sign up

Export Citation Format

Share Document