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.

Holomorphic Vanishing Theorems on Finsler Holomorphic Vector Bundles and Complex Finsler Manifolds

2018 ◽  
Vol 62 (3) ◽  
pp. 623-641
Author(s):  
Bin Shen
Keyword(s):  
Vector Fields ◽  
Vector Bundles ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Vanishing Theorems ◽  
Complex Manifolds ◽  
Finsler Manifolds ◽  
Holomorphic Vector Fields ◽  
Holomorphic Vector Bundles ◽  
Holomorphic Sections

AbstractIn this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.

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.

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 ASYMPTOTIC TORSION AND TOEPLITZ OPERATORS

2015 ◽  
Vol 16 (2) ◽  
pp. 223-349 ◽  
Author(s):  
Jean-Michel Bismut ◽  
Xiaonan Ma ◽  
Weiping Zhang
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Differential Form ◽  
Toeplitz Operators ◽  
Vector Bundles ◽  
Direct Image ◽  
Analytic Torsion ◽  
Leading Term ◽  
Positive Line Bundle ◽  
Positive Line

We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles $F_{p}$. For $p\in \mathbf{N}$, the flat vector bundle $F_{p}$ is the direct image of $L^{p}$, where $L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

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