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Rank-2 ample vector bundles on some smooth rational surfaces  [PDF]
Hironobu Ishihara
Mathematics , 1996,
Abstract: Some classification results for ample vector bundles of rank 2 on Hirzebruch surfaces, and on Del Pezzo surfaces, are obtained. In particular, we classify rank-2 ample vector bundles with $c_2$ less than 7 on Hirzebruch surfaces, and with $c_2$ less than 4 on Del Pezzo surfaces.
Ample vector bundles and branched coverings  [PDF]
Thomas Peternell,Andrew J. Sommese
Mathematics , 1999,
Abstract: Given a covering f: X \to Y of projective manifolds, we consider the vector bundle E on Y given as the dual of f_*(\O_X) / \O_Y. This vector bundles often has positivity properties, e.g. E is ample when Y is projective space by a theorem of Lazarsfeld. In general however E will not be ample due to the geometry of Y. We prove various results when E is spanned, nef or generically nef, under some assumptions on the base Y.
Nefness of adjoint bundles for ample vector bundles of corank 3  [PDF]
Andrea Luigi Tironi
Mathematics , 2011,
Abstract: Let $\mathcal{E}$ be an ample vector bundle of rank $r\geq 2$ on a smooth complex projective variety $X$ of dimension $n$. The aim of this paper is to describe the structure of pairs $(X,\mathcal{E})$ as above whose adjoint bundles $K_X+\det\mathcal{E}$ are not nef for $r=n-3$. Furthermore, we give some immediate consequences of this result in adjunction theory.
Nefness of adjoint bundles for ample vector bundles
Hidetoshi Maeda
Le Matematiche , 1995,
Abstract: Let E be an ample vector bundle of rank >1 on a smooth complex projective variety X of dimension n. This paper gives a classification of pairs (X,E) whose adjoint bundles K_X+det E are not nef in the case when r=n-2.
Positivity and vanishing theorems for ample vector bundles  [PDF]
Kefeng Liu,Xiaofeng Sun,Xiaokui Yang
Mathematics , 2010,
Abstract: In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact K\"ahler manifold $X$, $S^kE\ts \det E$ is both Nakano-positive and dual-Nakano-positive for any $k\geq 0$. Moreover, $H^{n,q}(X,S^kE\ts \det E)=H^{q,n}(X,S^kE\ts \det E)=0$ for any $q\geq 1$. In particular, if $(E,h)$ is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S^kE\ts \det E, S^kh\ts \det h)$ is both Nakano-positive and dual-Nakano-positive for any $k\geq 0$.
Vanishing theorems for ample vector bundles  [PDF]
Laurent Manivel
Mathematics , 1996, DOI: 10.1007/s002220050126
Abstract: We prove a general vanishing theorem for the cohomology of products of symmetric and skew-symmetric powers of an ample vector bundle on a smooth complex projective variety. Special cases include an extension of classical theorems of Griffiths and Le Potier to the whole Dolbeault cohomology, and an answer to a problem raised by Demailly. An application to degeneracy loci is given.
Parabolic k-ample bundles  [PDF]
Indranil Biswas,Fatima Laytimi
Mathematics , 2011,
Abstract: We construct projectivization of a parabolic vector bundle and a tautological line bundle over it. It is shown that a parabolic vector bundle is ample if and only if the tautological line bundle is ample. This allows us to generalize the notion of a k-ample bundle, introduced by Sommese, to the context of parabolic bundles. A parabolic vector bundle $E_*$ is defined to be k-ample if the tautological line bundle ${\mathcal O}_{{\mathbb P}(E_*)}(1)$ is $k$--ample. We establish some properties of parabolic k-ample bundles.
Stable ample 2-vector bundles on Hirzebruch surfaces  [PDF]
Alexandru Sterian
Mathematics , 2013,
Abstract: We discuss stability conditions for all rank-2 ample vector bundles on Hirzebruch surfaces with the second Chern class less than 7.
Vector bundles on projective varieties whose restriction to ample subvarieties split  [PDF]
Mihai Halic
Mathematics , 2014,
Abstract: We systematically study the splitting of vector bundles on a smooth, projective variety, whose restriction to the zero locus of a regular section of an ample vector bundle splits. First, we find ampleness and genericity conditions which ensure that the splitting of the vector bundle along the subvariety implies its global splitting. Second, we obtain a simple splitting criterion for vector bundles on the Grassmannian and on partial flag varieties.
Effective vanishing theorems for ample and globally generated vector bundles  [PDF]
Kefeng Liu,Xiaokui Yang
Mathematics , 2013,
Abstract: By proving an integral formula of the curvature tensor of $E\ts \det E$, we observe that the curvature tensor of $E\ts \det E$ is very similar to that of a line bundle and obtain certain new Kodaira-Akizuki-Nakano type vanishing theorems for vector bundles. As special cases, we deduce vanishing theorems for ample, nef and globally generated vector bundles by analytic method instead of the Leray-Borel-Le Potier spectral sequence.
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