On Frobenius and Fibers of Arithmetic Jet Spaces

Czechoslovak Mathematical Journal

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Mathematical Research Letters, 1996

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.

Annali di Matematica Pura ed Applicata, 1991

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.

Selecta Mathematica, 2011

We study derivations and differential forms on the arithmetic jet spaces of smooth schemes, relative to several primes. As applications we give a new interpretation of arithmetic Laplacians and we discuss the de Rham cohomology of some specific arithmetic jet spaces.

Journal of Algebra

If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.

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.

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

Journal of Pure and Applied Algebra, 1995

In this paper we compute the second homology of the discrete jet groups. Let R be the additive group of real numbers and R + the multiplicative group of positive reals. The n th jet group J n = {rx + a 2 x 2 + · · · + a n x n | r ∈ R + , a i ∈ R} is the group, under composition followed by truncation, of invertible, orientationpreserving real n-jets at 0. Consider the homomorphism D : J n → R + obtained by projecting onto the first coefficient, i.e. Df = first derivative of f at 0. Every jet with slope not equal to 1 is conjugate to its linear part. It follows there is a split exact sequence