scholarly journals Ample vector bundles on open algebraic varieties

1993 ◽  
Vol 29 (6) ◽  
pp. 885-910
Author(s):  
Shigeharu Takayama
Keyword(s):  
Vector Bundles ◽  
Algebraic Varieties

scholarly journals Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

2021 ◽  
Vol 20 (1) ◽  
pp. 95-119
Author(s):  
Nathan Grieve
Keyword(s):  
New York ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Index Theorem ◽  
Polynomial Functions ◽  
Algebraic Varieties ◽  
Wedderburn Decomposition ◽  
Generic Vanishing ◽  
Roch Theorem

Abstract We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A. Küronya and Y. Mustopa [Adv. Geom. 20 (2020), no. 3, 401-412]. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety’s endo-morphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that was established in N. Grieve [New York J. Math. 23 (2017), 1087-1110]. In a complementary direction, we explain how these topics pertain to the Index and Generic Vanishing Theory conditions for simple semihomogeneous vector bundles. In doing so, we refine results from M. Gulbrandsen [Matematiche (Catania) 63 (2008), no. 1, 123–137], N. Grieve [Internat. J. Math. 25 (2014), no. 4, 1450036, 31] and D. Mumford [Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100].

scholarly journals Complex vector bundles on real algebraic varieties of small dimension

1988 ◽  
Vol 38 (3) ◽  
pp. 345-349
Author(s):  
Wojciech Kucharz
Keyword(s):  
Algebraic Variety ◽  
Vector Bundles ◽  
Small Dimension ◽  
Real Algebraic Variety ◽  
Algebraic Varieties ◽  
Complex Vector ◽  
Real Algebraic Varieties ◽  
Continuous Complex

LetXbe an affine real algebraic variety. In this paper, assuming that dimX≤ 7 and thatXsatisfies some other reasonable conditions, we give a characterisation of those continuous complex vector bundles onXwhich are topologically isomorphic to algebraic complex vector bundles onX.

scholarly journals Positivity of vector bundles and Hodge theory

2021 ◽  
pp. 2140008
Author(s):  
Mark Green ◽  
Phillip Griffiths
Keyword(s):  
Differential Geometry ◽  
Vector Bundles ◽  
Special Property ◽  
Hodge Theory ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Curvature Invariants ◽  
Norm Property ◽  
Expository Paper

Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are [Formula: see text] order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property.

scholarly journals NC-smooth algebroid thickenings for families of vector bundles and quiver representations

2019 ◽  
Vol 155 (4) ◽  
pp. 681-710
Author(s):  
Ben Dyer ◽  
Alexander Polishchuk
Keyword(s):  
Deformation Quantization ◽  
Vector Bundles ◽  
Algebraic Varieties ◽  
Quiver Representations ◽  
Projective Curves

In his work on deformation quantization of algebraic varieties Kontsevich introduced the notion ofalgebroidas a certain generalization of a sheaf of algebras. We construct algebroids which are given locally by NC-smooth thickenings in the sense of Kapranov, over two classes of smooth varieties: the bases of miniversal families of vector bundles on projective curves, and the bases of miniversal families of quiver representations.

STABLE ULRICH BUNDLES

2012 ◽  
Vol 23 (08) ◽  
pp. 1250083 ◽  
Author(s):  
MARTA CASANELLAS ◽  
ROBIN HARTSHORNE ◽  
FLORIAN GEISS ◽  
FRANK-OLAF SCHREYER
Keyword(s):  
Vector Bundles ◽  
Sufficient Conditions ◽  
Algebraic Varieties ◽  
Stable Bundles ◽  
Cubic Surfaces ◽  
Number Of Generators ◽  
Class D ◽  
First Chern Class ◽  
Necessary And Sufficient ◽  
Acm Bundles

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces X ⊂ ℙ3. We give necessary and sufficient conditions on the first Chern class D for the existence of stable Ulrich bundles on X of rank r and c1 = D. When such bundles exist, we prove that the corresponding moduli space of stable bundles is smooth and irreducible of dimension D2 - 2r2 + 1 and consists entirely of stable Ulrich bundles (see Theorem 1.1). We are also able to prove the existence of stable Ulrich bundles of any rank on nonsingular cubic threefolds in ℙ4, and we show that the restriction map from bundles on the threefold to bundles on the surface is generically étale and dominant.

The loops onU(n)/O(n) andU(2n)/Sp(n)

1988 ◽  
Vol 104 (1) ◽  
pp. 95-103 ◽  
Author(s):  
M. C. Crabb ◽  
S. A. Mitchell
Keyword(s):  
Vector Bundles ◽  
Geometric Realization ◽  
The Other ◽  
Complex Case ◽  
Integral Homology ◽  
Algebraic Varieties ◽  
The Real ◽  
Symmetric Powers ◽  
Precise Statement ◽  
Real Algebraic Varieties

In [6] and [9] the second author and Bill Richter showed that the natural ‘degree’ filtration on the homology of ΩSU(n) has a geometric realization, and that this filtration stably splits (as conjectured by M. Hopkins and M. Mahowald). The purpose of the present paper is to prove the real and quaternionic analogues of these results. To explain what this means, consider the following two ways of viewing the filtration and splitting for ΩSU(n). Whenn= ∞, ΩSU=BU. The filtration isBU(1)⊆BU(2)⊆… and the splittingBU≅ V1≤<∞is a theorem of Snaith[14]. The result for ΩSU(n) may then be viewed as a ‘restriction’ of the result forBU. On the other hand there is a well-known inclusion ℂPn−1. This extends to a map ΩΣℂPn−1→ΩSU(n), and the filtration (or splitting) may be viewed, at least algebraically, as a ‘quotient’ of the James filtration (or splitting) of ΩΣℂPn−1. It is now clear what is meant by the ‘real and quaternionic analogues’. In the quaternionic case, we replaceBUbyBSp=Ω(SU/SP), ΩSU(n) by Ω(SU(2n)/SP(n))and ℂPn−1by ℍPn−1. The integral homology of Ω(SU(2n)/SP(n)) is the symmetric algebra on the homology of ℍPn−1, and may be filtered by the various symmetric powers. We show that this filtration can be realized geometrically, and that the spaces of the filtration are certain (singular) real algebraic varieties (exactly as in the complex case). The strata of the filtration are vector bundles over filtrations of Ω(SU(2n−2)/SP(n−1)), and the filtration stably splits. See Theorems (1·7) and (2·1) for the precise statement. In the real case we replaceBUby Ω(SU/SO), Ω(SU(n)/SO(n)) and ℂPn−1by ℝPn−1. Here integral homology must be replaced by mod 2 homology, and splitting is only obtained after localization at 2. (Snaith's splitting ofBOin [14] can be refined [2, 8] so as to be exactly analogous to the splitting ofBU:BO≅V1≤<∞MO(k).)

scholarly journals Realization of Chern classes by subvarieties with certain singularities

1980 ◽  
Vol 80 ◽  
pp. 49-74 ◽  
Author(s):  
Hiroshi Morimoto
Keyword(s):  
Vector Bundles ◽  
Chern Classes ◽  
Algebraic Varieties ◽  
Stein Manifolds ◽  
Holomorphic Vector Bundles ◽  
Definition Of

In this paper we are concerned with subvarieties which realize Chern classes of holomorphic vector bundles. The existence of these subvarieties is known in some cases (for instance, see A. Grotheridieck [2] for projective algebraic varieties and M. Cornalba and P. Griffiths [1] for Stein manifolds). In the present paper we realize Chern classes by subvarieties with singularities of a certain type. Our main theorem is as follows (see Def. 1.1.3 for the definition of quasilinear subvarieties).

Vector bundles over real algebraic varieties

K-Theory ◽  
1989 ◽  
Vol 3 (3) ◽  
pp. 271-298 ◽  
Author(s):  
J. Bochnak ◽  
M. Buchner ◽  
W. Kucharz
Keyword(s):  
Vector Bundles ◽  
Algebraic Varieties ◽  
Real Algebraic Varieties
Sign in / Sign up

Export Citation Format

Share Document