scholarly journals Nef vector bundles on a projective space or a hyperquadric with the first Chern class small

2021 ◽  
Author(s):  
Masahiro Ohno
Keyword(s):  
Projective Space ◽  
Chern Class ◽  
Vector Bundles ◽  
First Chern Class ◽  
Nef Vector Bundles

Nef vector bundles on a quadric surface with the first Chern class (2, 1)

2020 ◽  
Vol 20 (1) ◽  
pp. 109-116
Author(s):  
Masahiro Ohno
Keyword(s):  
Chern Class ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Closed Field ◽  
Quadric Surface ◽  
Algebraically Closed Field ◽  
First Chern Class ◽  
Smooth Quadric Surface ◽  
Nef Vector Bundles

AbstractWe classify nef vector bundles on a smooth quadric surface with the first Chern class (2, 1) over an algebraically closed field of characteristic zero; we see in particular that such nef bundles are globally generated.

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 On the positivity of the first Chern class of an Ulrich vector bundle

2021 ◽  
pp. 2150071
Author(s):  
Angelo Felice Lopez
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Vector Bundles ◽  
First Chern Class

We study the positivity of the first Chern class of a rank [Formula: see text] Ulrich vector bundle [Formula: see text] on a smooth [Formula: see text]-dimensional variety [Formula: see text]. We prove that [Formula: see text] is very positive on every subvariety not contained in the union of lines in [Formula: see text]. In particular, if [Formula: see text] is not covered by lines we have that [Formula: see text] is big and [Formula: see text]. Moreover we classify rank [Formula: see text] Ulrich vector bundles [Formula: see text] with [Formula: see text] on surfaces and with [Formula: see text] or [Formula: see text] on threefolds (with some exceptions).

scholarly journals BRILL–NOETHER THEOREMS AND GLOBALLY GENERATED VECTOR BUNDLES ON HIRZEBRUCH SURFACES

2018 ◽  
Vol 238 ◽  
pp. 1-36 ◽  
Author(s):  
IZZET COSKUN ◽  
JACK HUIZENGA
Keyword(s):  
Projective Plane ◽  
Euler Characteristic ◽  
Chern Class ◽  
Vector Bundles ◽  
Betti Numbers ◽  
Stable Bundle ◽  
Hirzebruch Surface ◽  
Hirzebruch Surfaces ◽  
Chern Characters ◽  
First Chern Class

In this paper, we show that the cohomology of a general stable bundle on a Hirzebruch surface is determined by the Euler characteristic provided that the first Chern class satisfies necessary intersection conditions. More generally, we compute the Betti numbers of a general stable bundle. We also show that a general stable bundle on a Hirzebruch surface has a special resolution generalizing the Gaeta resolution on the projective plane. As a consequence of these results, we classify Chern characters such that the general stable bundle is globally generated.

scholarly journals Stable vector bundles on an algebraic surface

1975 ◽  
Vol 58 ◽  
pp. 25-68 ◽  
Author(s):  
Masaki Maruyama
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Algebraic Curve ◽  
Algebraic Surface ◽  
Vector Bundles ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Stable Vector Bundles ◽  
First Chern Class

Let X be a non-singular projective algebraic curve over an algebraically closed field k. D. Mumford introduced the notion of stable vector bundles on X as follows;DEFINITION ([7]). A vector bundle E on X is stable if and only if for any non-trivial quotient bundle F of E,where deg ( • ) denotes the degree of the first Chern class of a vector bundles and r( • ) denotes the rank of a vector bundle.

scholarly journals GAUGE-EQUIVARIANT HILBERT BIMODULES AND CROSSED PRODUCTS BY ENDOMORPHISMS

2009 ◽  
Vol 20 (11) ◽  
pp. 1363-1396 ◽  
Author(s):  
EZIO VASSELLI
Keyword(s):  
Vector Bundle ◽  
Moduli Space ◽  
Chern Class ◽  
Vector Bundles ◽  
Crossed Products ◽  
Tensor Categories ◽  
Dual Object ◽  
Nontrivial Center ◽  
First Chern Class ◽  
The Given

C*-endomorphisms arising from superselection structures with nontrivial center define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz–Pimsner algebra of a vector bundle having the above-mentioned rank, first Chern class and can be used to construct a duality for abstract (nonsymmetric) tensor categories versus group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e. the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with the given endomorphism. The above-mentioned Hilbert bimodules are noncommutative analogs of gauge-equivariant vector bundles in the sense of Nistor–Troitsky.

scholarly journals The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

2021 ◽  
Vol 9 ◽  
Author(s):  
Patrick Graf ◽  
Martin Schwald
Keyword(s):  
Chern Class ◽  
Cohomology Group ◽  
Canonical Bundle ◽  
Deformation Space ◽  
Projective Varieties ◽  
Quotient Singularities ◽  
Finite Quotient ◽  
First Chern Class ◽  
Small Deformations

Abstract Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

scholarly journals ON THE STABILITY OF SYZYGY BUNDLES

2011 ◽  
Vol 22 (04) ◽  
pp. 515-534 ◽  
Author(s):  
IUSTIN COANDĂ
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Vector Bundles ◽  
Elementary Proof ◽  
Base Point ◽  
Vector Spaces ◽  
Similar Argument ◽  
Vanishing Theorem ◽  
The Stability ◽  
Koszul Cohomology

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

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