scholarly journals The stability of certain vector bundles on multi-projective spaces

2007 ◽  
Vol 2 ◽  
pp. 3085-3088
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundles ◽  
Projective Spaces ◽  
The Stability

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)].

Hermitian–Einstein metrics and jumping lines forSp(n+1)-homogeneous bundles over ℙ2n+1

1993 ◽  
Vol 114 (3) ◽  
pp. 443-451
Author(s):  
Al Vitter
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Vector Bundles ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Holomorphic Vector Bundles ◽  
Kähler Form ◽  
Points Of View ◽  
The Stability ◽  
Complex Projective

Stable holomorphic vector bundles over complex projective space ℙnhave been studied from both the differential-geometric and the algebraic-geometric points of view.On the differential-geometric side, the stability ofE-→ ℙncan be characterized by the existence of a unique hermitian–Einstein metric onE, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn(see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙnand the nullcorrelation bundle (see [9] and [10]).

scholarly journals Torus-equivariant vector bundles on projective spaces

1988 ◽  
Vol 111 ◽  
pp. 25-40 ◽  
Author(s):  
Tamafumi Kaneyama
Keyword(s):  
Vector Bundle ◽  
Toric Variety ◽  
Vector Bundles ◽  
Rational Point ◽  
Projective Spaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Algebraic Torus

For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

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