Stability and cohomology of kernel bundles on projective space (original) (raw)

Cohomological characterisation of Steiner bundles

Forum Mathematicum, 2009

A vector bundle E on a smooth irreducible algebraic variety X is called a Steiner bundle of type (F 0 , F 1) if it is defined by an exact sequence of the form 0 → F s 0 ϕ − → F t 1 → E → 0, where s, t ≥ 1 and (F 0 , F 1) is a strongly exceptional pair of vector bundles on X such that F ∨ 0 ⊗ F 1 is generated by global sections. Let X be a smooth irreducible projective variety of dimension n with an n-block collection B = (E 0 , E 1 ,. .. , E n), E i = (E i 1 ,. .. , E i α i), of locally free sheaves on X which generate D b (O X −mod). We give a cohomological characterisation of Steiner bundles of type (E a i 0 , E b j 0) on X , where 0 ≤ a < b ≤ n and 1 ≤ i 0 ≤ α a , 1 ≤ j 0 ≤ α b .

On the Stability of Homogeneous Vector Bundles

Journal of Mathematical Sciences the University of Tokyo, 2004

Let G be a connected semisimple linear algebraic group over an algebraically closed field k and P ⊂ G a parabolic subgroup without any simple factor. Let V be an irreducible P-module and E P (V) = (G × V)/P the associated vector bundle over G/P. We prove that E P (V) is stable with respect to any polarization on G/P. In [Um] this was proved under the assumption that the characteristic of k is zero and the question was asked whether it remains valid when the characteristic is positive.

The stable geometric dimension of vector bundles over real projective spaces

Transactions of the American Mathematical Society, 1981

An elementary argument shows that the geometric dimension of any vector bundle of order 2e over RP" depends only on e and the residue of n mod 8 for n sufficiently large. In this paper we calculate this geometric dimension, which is approximately le. The nonlifting results are easily obtained using the spectrum bJ. The lifting results require fto-resolutions. Half of the paper is devoted to proving Mahowald's theorem that beginning with the second stage io-resolutions act almost like ^(Z^-resolutions.

On the classification of complex vector bundles of stable rank

2006

One describes, using a detailed analysis of Atiyah-Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constructive terms the Chern classes of a vector bundle.

Stability of projective Poincare and Picard bundles

Bulletin of the London Mathematical Society, 2009

Let X be an irreducible smooth projective curve of genus g ≥ 3 defined over the complex numbers and let M ξ denote the moduli space of stable vector bundles on X of rank n and determinant ξ, where ξ is a fixed line bundle of degree d. If n and d have a common divisor, there is no universal vector bundle on X × M ξ. We prove that there is a projective bundle on X × M ξ with the property that its restriction to X × {E} is isomorphic to P (E) for all E ∈ M ξ and that this bundle (called the projective Poincaré bundle) is stable with respect to any polarization; moreover its restriction to {x} × M ξ is also stable for any x ∈ X. We prove also stability results for bundles induced from the projective Poincaré bundle by homomorphisms PGL(n) → H for any reductive H. We show further that there is a projective Picard bundle on a certain open subset M ′ of M ξ for any d > n(g − 1) and that this bundle is also stable. We obtain new results on the stability of the Picard bundle even when n and d are coprime.