Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology. On certain extensions of vector bundles in p adic geometry 3 theorem 1. Christopher deninger and annette werner have constructed a functorial parallel transport for vector bundles on adic varieties satisfying a. All structured data from the file and property namespaces is available under the creative commons cc0 license. In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space x for example x could be a topological space, a manifold, or an algebraic variety. Vector bundles and padic galois representations institute. On tannaka duality for vector bundles on padic curves 5 12. In particular, fargues and fontaine proved that any bundle is determined up to isomorphism by. In the case there is no ambiguity about the base, we will denote a vector bundle by efor short. F now also refered to as the fundamental curve of p adic hodge theory, study its properties and classify vector bundles on it. Vector bundles over an elliptic curve 417 it is almost immediate that the map. Homogeneous padic vector bundles on abelian varieties that. One then takes the associated quasicoherent sheaf on x. The notion of vector bundle is a basic extension to the geometric domain of the fundamental idea of a vector space.
Chapter 7 vector bundles louisiana state university. Files are available under licenses specified on their description page. For any smooth manifold m, e m rk is a trivial bundle over m. An holomorphic hequivariant vector bundle over x fe is a vector bundle equipped with a cont. Peter scholzes lectures on padic geometry, fall 2014. In the usual terminology w is the universal bundle over the classifying. It wont be the case for the curves we are going to construct. The only two vector bundles with base space a circle and onedimensional.
I am currently an nsf postdoc at stanford university, working with ravi vakil. A section of a tangent vector bundle is a vector field. An introduction to the topology of the moduli space of. On tannaka duality for vector bundles on padic curves. Another example is the fouriersato transform, defined on the category of conical sheaves over a fixed real vector bundle. We are investigating homogeneous p adic vector bundles on abelian varieties that are analytic tori. They show that these vector bundles are closely related to p adic. Recall that a vector bundle on a smooth projective curve over. Let f be a perfect field of characteristic p 0 complete. This is sometimes called the fundamental exact sequence of padic hodge theory. For these bundles it was possible to establish a partial padic analogue of the classical narasimhan seshadri theory for semistable vector bundles of degree zero on compact riemann surfaces. They show that these vector bundles are closely related to padic representations of g k, where kis any nite extension of q p. X be a vector bundle of rank r over an algebraic variety x.
Special thanks to brian conrad for his thorough comments on every lecture. In view of the results 4 one might hope that every numerically. When f is arbitrary but the vector bundle with connection in question is a nontrivial line bundle, we use techniques from the theory of arithmetic dmodules such as frobenius descent, micro. E induces an equivalence of categories betweensemistable vector bundles over x of slope 0 and nite dimensonal q p vector spaces. Given some context of geometry, then a vector bundle is a collection of vector spaces that varies in a geometric way over a given base space x x. We develop a theory of etale parallel transport for vector bundles with numerically flat reduction on a padic variety. The proof combines a dynamical argument on banachcolmez spaces with an optimal existence theorem. This is a revised version of the lecture notes for the course on padic geometry given by p. Thanks to arthur ogus and jay pottharst for helpful comments. Kequivariant vector bundles over x, we recover classical p adic hodge theory as a special case.
A section is an abstract characterization of what it means to be a graph. Werner, vector bundles and p adic representations i, preprint, 2003. Mar 04, 2011 the present paper is concerned with the algebraic feature of double vector bundles. The most important example is the tangent bundle, a real vector bundle. Ag 14 jun 2005 on tannaka duality for vector bundles on christopher deninger annette werner padic curves dedicated to jacob murre 1 introduction in our paper dw2 we have introduced a certain category bps of degree zero bundles with potentially strongly semistable reduction on a padic curve. A vector bundle of rank 1 is usually called a line bundle. Choose from over a million free vectors, clipart graphics, vector art images, design templates, and illustrations created by artists worldwide. C of complex valued functions on a nitedimensional vector space lover k. Seminar vector bundles on curves and padic hodge theory. Vector bundles and padic galois representations imjprg. Moreover, every stable bundle of degree zero comes from an irreducible unitary representation. On tannaka duality for vector bundles on adic curves.
We have developed the notion of a cp vector bundle over xas a certain kind of cp mapping e xthat is roughly speaking a cp. E is obtained from the cartesian product g x e by identifying gu x e with g x u. In particular, a vector field on a smooth manifold. But it is not clear from this description that this is a vector bundle. But there are many other interesting vector bundles. Crew, universal extensions and padic periods of elliptic curves, comp. Pdf vector bundles on padic curves and parallel transport. Mark reeder padic whittaker functions and vector bundles on flag manifolds. E, and fbe vector bundles on xsuch that the maximum slope in hnd is less than the minimum slope in hnf. Vector bundles on padic curves and parallel transport. Christopher deninger and annette werner have constructed a functorial parallel transport for vector bundles on adic varieties satisfying a condition on their reduction. One of the main open questions of dw2 was, whether our category was abelian. Parallel transport for vector bundles on padic varieties.
A kvector bundle over mof rank kconsists of a bundle e m whose. We can now state the classi cation theorem for vector bundles on xas follows. Lvector spaces equipped with bijective semilinear frobenius endomorphism. A vector bundle is semistable of slope if and ony if eis isomorphic to a direct sum of copies of o x. Homogeneous vector bundles raoul bott the annals of. Vector bundles and padic galois representations 3 remark. Such correspondences arise in the context of padic simpson theory in the case of vanishing higgs. Stable bundles there are at least two ways around the problem. Local systems on diamonds and padic vector bundles lucas mann and annette werner may 15, 2020 we use scholzes framework of diamonds to gain new insights in correspondences between padic vector bundles and local systems.
Homogeneous padic vector bundles on abelian varieties. Strictly speaking, a vector bundle is an entire triple e. On tannaka duality for vector bundles on adic curves arxiv. An algebraic vector bundle of rank r on x is a linear. I received my phd in june 2019 from mit, where i was a student of bjorn poonen and joe harris. We have developed the notion of a cp vector bundle over xas a certain kind of cp mapping e xthat is roughly speaking a cp varying family of. Degenerating vector bundles in p adic hodge theory david hansen. A topological ring r is called a tate ring if it is fadic and contains a topologically nilpotent unit. We show that for each homogeneous vector bundle on such a variety there exists an integer n 0, such that the pullback of this vector bundle via the nmultiplication is attached to an integral representation of the topological fundamental group. Unfortunately we do not have a padic banach version of the weil representation on a padic vector space, and this approach cannot be taken. All bundles of degree zero whose reduction is strongly semistable belong to this class.
The weil representation 17 over a nite eld kis the algebra object that governs the symmetries of the standard hilbert space h l l2l. Further, the umodule e defines an analytic vector bundle e g x e over x, the author holds a sloan fellowship. Vector bundles and connections universiteit utrecht. X which is locally trivial, that is, for any x 2 x there exists an open neighborhood u of x and an isomorphism of. Abstract we define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p adic curve. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. In the context of the padic langlands program, it is more natural to consider the mod pof a unitary banach representation than the mod pof a classical admissible representation of a padic group. Motivation let x,o be a cp premanifold with corners, 0. Download free vectors, clipart graphics, vector art. In this paper we will be interested only with the case where kis of. Linear algebra operations on vector bundles motivation.
I have reformatted this with narrower margins for a better reading experience on devices like an ipad, but for a paper copy with more standard size margins try printing at 8590 per cent of full size. Vector bundles and structure groups a vector bundle over a topological space m or with base space m is, essentially, family of vector spaces continuously parametrized by m. E f is the tensor product bundle, which has rank rk erk f. We are investigating homogeneous padic vector bundles on abelian varieties that are analytic tori. They show that these vector bundles are closely related to p adic representations of g k, where kis any nite extension of q p.
As usual, if x is a noetherian scheme, we view a vector bundle over x as a locally free coherent o xmodule. The present paper establishes a partial p adic analogue of this theory, generalized to representations of the fundamental groupoid. August 22, 2018 abstract we compute the closure relations among the individual hardernarasimhan strata in the moduli stack of rank n vector bundles on the farguesfontaine curve. Degenerating vector bundles in padic hodge theory david hansen. One of the main results of ff is a complete classi cation of vector bundles on x. This new vector bundle fe is called the \associated vector bundle to e, given by the functor f.
Beginning in september 2020, i will be an assistant professor at university of washington at seattle my research is in the intersection of algebraic geometry and number theory. The difference is that, for a vector bundle, there is usually no natural lie group action on the total space that acts transitively on the fibers. A bundle over a manifold is trivial if it is simply the cartesian product of. Im using the letter m to denote the base space of the vector bundle as a concession to the fact that in most of the applications well be. The present paper establishes a partial padic analogue of this. The degree of a vector bundle eof rank ranke r 1 is the degree of re. In the context of the p adic langlands program, it is more natural to consider the mod pof a unitary banach representation than the mod pof a classical admissible representation of a p adic group. When f is arbitrary but the vector bundle with connection in question is a nontrivial line bundle, we use techniques from the theory of arithmetic dmodules. Peter scholzes lectures on padic geometry, fall 2014 last updated on 29 jan. The present paper establishes a partial padic analogue of this theory, generalized to representations of the fundamental groupoid. Vector bundles and padic galois representations laurent fargues and jeanmarc fontaine abstract. The p p adic hodge structure of a proper or semistably compactified padic analytic variety is essentially given by a relation between three important invariants of the given variety. V defined by assigning to each x the subspace e x of v or equivalently the quotient space e x of v is such that1l 2, where11 denotes the sequence on x induced from 1 by. Unfortunately we do not have a p adic banach version of the weil representation on a p adic vector space, and this approach cannot be taken.
E induces an equivalence of categories betweensemistable vector bundles over x of slope 0 and nite dimensonal q pvector spaces. Given a space x, we take a real or complex finite dimensional vector space v and. Then there exists a short exact sequence of vector bundles on x 0. One is to replace the notion of a moduli space with that of a moduli stack. If a group g acts on the left on a noetherian scheme x, an o xrepresentation. Introduction in this note we study the deformation properties of algebraic cycle classes modulo rational equivalence.
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