A remark on the jet bundles over the projective line (original) (raw)

Line bundles for which a projectivized jet bundle is a product

1998

We characterize the triples (X,L,H), consisting of holomorphic line bundles L and H on a complex projective manifold X, such that for some positive integer k, the k-th holomorphic jet bundle of L, J_k(L), is isomorphic to a direct sum H+...+H. Given the geometrical constrains imposed by a projectivized line bundle being a product of the base and a projective space it is natural to expect that this would happen only under very rare circumstances. It is shown, in fact, that X is either an Abelian variety or projective space. In the former case L\cong H is any line bundle of Chern class zero. In the later case for k a positive integer, L=O_{P^n}(q) with J_k(L)=H+...+H if and only if H=O_{P^n}(q-k) and either q\ge k or q\le -1.

Jets of line bundles on curves and Wronskians

Journal of Pure and Applied Algebra, 2011

We collect a few results about jets of line bundles on curves and Wronskians, with a special emphasis to those arising from the canonical involution of a hyperelliptic curve.

Open questions and recent results in the theory of geometrized first-order jet bundles

Within the framework of first order jet-Generalized Lagrange Spaces, the present survey article provides a survey presenting the work of a Romanian research group in the field of d−geometric structures on the first order jet space J 1 (T, M). Recent advances and actual open questions regarding these basic distinguished structures -which extend the corresponding generalized Lagrange, Lagrange, Finsler and Riemann d−structures of the tangent bundle framework, are described.

On finite generation of fiber ring of invariant jet differentials

Cornell University - arXiv, 2020

We prove that the fiber ring of the space of invariant jet differentials of a projective manifold is finitely generated on the regular locus. Berczi-Kirwan has partially worked out the question in [2]; however, our method is different and complementary. The analytic automorphism group of regular k-jets of holomorphic curves on a projective variety X is a non-reductive subgroup of the general linear group GL k C. In this case, the Chevalley theorem on the invariant polynomials in the fiber rings fails in general. Thus, the analysis of Cartan subalgebras of the Lie algebra and its Weyl group requires different methods. We employ some techniques of algebraic Lie groups (not necessarily reductive) together with basic results obtained in [2] to prove the finite generation of the stalk ring at a regular point.

Algebraic Characterization of Affine Structure on Jet and Weil Bundles ∗

2005

We characterize natural transformation between Weil Bundles that are endowed with a canonical affine structure and show several cases. Those transformations are often passed down to Jet Spaces, and we characterize the cases in which the affine structure is also passed down. We prove that the classical situation is an example and give some practical generalization.

Invariant Jet differentials and Asymptotic Serre duality

Cornell University - arXiv, 2020

We generalize the main result of Demailly [3] for the bundles E GG k,m (V *) of jet differentials of order k and weighted degree m to the bundles E k,m (V *) of the invariant jet differentials of order k and weighted degree m. Namely, Theorem 0.5 from [3] and Theorem 9.3 from [2] provide a lower bound c k k m n+kr−1 on the number of the linearly independent holomorphic global sections of E GG k,m V * O(−mδA) for some ample divisor A. The group G k of local reparametrizations of (C,0) acts on the k-jets by orbits of dimension k, so that there is an automatic lower bound c k k m n+kr−1 on the number of the linearly independent holomorphic global sections of E k,m V * O(−mδA). We formulate and prove the existence of an asymptotic duality along the fibers of the Green-Griffiths jet bundles over projective manifolds. We also prove a Serre duality for asymptotic sections of jet bundles. An application is also given for partial application to the Green-Griffiths conjecture.

On the local structure of A-jet spaces 1

2001

We analyze the local structure of A-jet spaces, where A is a Weil algebra;by the way, we introduce the bundles of A-jets of sections of a regular projection and describe their vertical tangent spaces.

Canonical involution on double jet bundles

TURKISH JOURNAL OF MATHEMATICS

In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by primary and secondary structures belong to the same atlas. We prove that double jet bundles can be considered as a quotient of second order jet bundle. We show that there exists a natural involution that interchanges between primary and secondary vector bundle structures on double jet bundles.

Affine structures on jet and Weil bundles

Colloquium Mathematicum, 2009

Weil algebra morphism induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cases, this structure of affine bundle is passed down to Jet spaces. We give a characterization of this fact in algebraic terms. This algebraic condition also determines an affine structure between the groups of automorphisms of related Weil algebras.

New Geometrical Objects on Jet Fibre Bundle of Order One

2000

Section 1 presents the main properties of the differentiable structure of the jet fibre bundle of order one J(T, M). Section 2 introduces an important collection of geometrical objects on J(T, M) as the d-tensors, the temporal and spatial sprays and the harmonic maps induced by these sprays. Moreover, we show that the notion of harmonic map induced by the sprays is a natural generalization of the classical notion of harmonic map between two Riemannian manifolds. In Section 3 we present the connection between the temporal and spatial sprays and the important notion of nonlinear connection on J(T, M). Section 4 studies the problem of prolongation of vector fields from T ×M to 1-jet space J(T, M). Mathematics Subject Classification: 53C07, 53C43, 53C99