Restriction properties for the Krull–Schmidt decomposition of vector bundles

2018 ◽  
Vol 29 (03) ◽  
pp. 1850022
Author(s):  
Mihai Halic
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
The Other ◽  
Flag Varieties ◽  
Schmidt Decomposition ◽  
Ample Vector Bundle ◽  
Projective Varieties ◽  
Splitting Criterion ◽  
Irreducible Components ◽  
Smooth Projective Varieties

We obtain decomposability criteria for vector bundles on smooth projective varieties [Formula: see text] by comparing the Krull–Schmidt decomposition on [Formula: see text], on one hand, and along the vanishing locus of a section in an ample vector bundle over [Formula: see text], on the other hand. We determine effective bounds for the amplitude of and also genericity conditions for its sections which ensure that the irreducible components of and those of its restriction correspond bijectively. Moreover, we get a simple splitting criterion for vector bundles on partial flag varieties.

Special varieties in adjunction theory and ample vector bundles

2001 ◽  
Vol 130 (1) ◽  
pp. 61-75 ◽  
Author(s):  
A. LANTERI ◽  
H. MAEDA
Keyword(s):  
Vector Bundles ◽  
Projective Variety ◽  
Global Section ◽  
Smooth Projective Variety ◽  
Ample Vector Bundle ◽  
Projective Varieties ◽  
Adjunction Theory ◽  
Set Up ◽  
Important Position ◽  
Smooth Projective Varieties

In this paper varieties are always assumed to be defined over the field [Copf ] of complex numbers.Given a smooth projective variety Z, the classification of smooth projective varieties X containing Z as an ample divisor occupies an extremely important position in the theory of polarized varieties and it is well-known that the structure of Z imposes severe restrictions on that of X. Inspired by this philosophy, we set up the following condition ([midast ]) in [LM1] in order to generalize several results on ample divisors to ample vector bundles:([midast ]) [Escr ] is an ample vector bundle of rank r [ges ] 2 on a smooth projective variety X of dimension n such that there exists a global section s ∈ Γ([Escr ]) whose zero locus Z = (s)0 is a smooth subvariety of X of dimension n − r [ges ] 1.

Ample Vector Bundles of Curve Genus One

1999 ◽  
Vol 42 (2) ◽  
pp. 209-213 ◽  
Author(s):  
Antonio Lanteri ◽  
Hidetoshi Maeda
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Ample Vector Bundle ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Curve Genus ◽  
Genus One ◽  
Complex Projective ◽  
Zero Locus

AbstractWe investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.

scholarly journals A Riemann–Hurwitz Theorem for the Algebraic Euler Characteristic

2017 ◽  
Vol 60 (3) ◽  
pp. 490-509
Author(s):  
Andrew Fiori
Keyword(s):  
Euler Characteristic ◽  
Euler Characteristics ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Irreducible Components ◽  
Arbitrary Dimensions ◽  
Smooth Projective Varieties ◽  
Ramification Locus ◽  
Hurwitz Theorem

AbstractWe prove an analogue of the Riemann–Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

scholarly journals Some Adjunction Properties of Ample Vector Bundles

2001 ◽  
Vol 44 (4) ◽  
pp. 452-458
Author(s):  
Hironobu Ishihara
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Sectional Genus ◽  
Genus Zero ◽  
Ample Vector Bundle

AbstractLet ε be an ample vector bundle of rank r on a projective variety X with only log-terminal singularities. We consider the nefness of adjoint divisors KX +(t−r) det ε when t ≥ dim X and t > r. As an application, we classify pairs (X, ε) with cr-sectional genus zero.

AMPLE VECTOR BUNDLES WITH SECTIONS VANISHING ON PROJECTIVE SPACES OR QUADRICS

1995 ◽  
Vol 06 (04) ◽  
pp. 587-600 ◽  
Author(s):  
ANTONIO LANTERI ◽  
HIDETOSHI MAEDA
Keyword(s):  
Vector Bundle ◽  
Projective Space ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Projective Spaces ◽  
Compact Complex Manifold ◽  
Ample Vector Bundle ◽  
Zero Locus

Let ɛ be an ample vector bundle of rank r≥2 on a compact complex manifold X of dimension n≥r+1 having a section whose zero locus is a submanifold Z of the expected dimension n–r. Pairs (X, ɛ) as above are classified under the assumption that Z is either a projective space or a quadric.

scholarly journals A Vanishing Theorem

2005 ◽  
Vol 180 ◽  
pp. 35-43 ◽  
Author(s):  
F. Laytimi ◽  
W. Nahm
Keyword(s):  
Vector Bundle ◽  
Tensor Product ◽  
Vector Bundles ◽  
Vanishing Theorem ◽  
Dolbeault Cohomology ◽  
Ample Vector Bundle ◽  
Exterior Powers ◽  
Parameter Values

AbstractThe main result is a general vanishing theorem for the Dolbeault cohomology of an ample vector bundle obtained as a tensor product of exterior powers of some vector bundles. It is also shown that the conditions for the vanishing given by this theorem are optimal for some parameter values.

scholarly journals P-Ample Bundles and Their Chern Classes

1971 ◽  
Vol 43 ◽  
pp. 91-116 ◽  
Author(s):  
David Gieseker
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Line Bundles ◽  
Chern Classes ◽  
Ground Field ◽  
Ample Vector Bundle

In [9], Hartshorne extended the concept of ampleness from line bundles to vector bundles. At that time, he conjectured that the appropriate Chern classes of an ample vector bundle were positive, and it was hoped that there would be some criterion for ampleness of vector bundles similar to Nakai’s criterion for line bundles. In the same paper, Hartshorne also introduced the notion of p-ample when the ground field had characteristic p, proved that a p-ample bundle was ample and asked if the converse were true.

Topologically trivial holomorphic vector bundles on infinite-dimensional projective varieties

2004 ◽  
Vol 134 (1) ◽  
pp. 33-38
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Finite Rank ◽  
Closed Subset ◽  
Vector Bundles ◽  
Complex Space ◽  
Holomorphic Vector Bundle ◽  
Projective Varieties ◽  
Infinite Dimensional ◽  
Holomorphic Vector Bundles ◽  
Locally Convex

Let V be an infinite-dimensional locally convex complex space, X a closed subset of P(V) defined by finitely many continuous homogeneous equations and E a holomorphic vector bundle on X with finite rank. Here we show that E is holomorphically trivial if it is topologically trivial and spanned by its global sections and in a few other cases.

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