The Hasse Invariant of a Vector Bundle

In this paper we construct Stiefel-Whitney and Euler classes in Chow cohomology for algebraic vector bundles with nondegenerate quadratic form. These classes are not in the algebra generated by the Chern classes of such bundles and are new characteristic classes in algebraic geometry. On complex varieties, they correspond to classes with the same name pulled back from the cohomology of the classifying space BSO(N,C). The classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups ([T2], [EG]). We begin by using the geometry of quadric bundles to study Chern classes of maximal isotropic subbundles. If V → X is a vector bundle with quadratic form, and if E and F are maximal isotropic subbundles of V then we prove (Theorem 1) that ci(E) and ci(F ) are equal mod 2. Moreover, if the rank of V is 2n, then cn(E) = ±cn(F ), proving a conjecture of Fulton. We define Stiefel-Whitney and Euler classes as Chow cohomology classes which pull back to Chern classes of maximal isotropic subbundles of the pullback bundle. Using the above theorem we show (Theorem 2) that these classes exist and are unique, even though V need not have a maximal isotropic subbundle. These constructions also make it possible to give “Schubert” presentations,

We introduce the notion of hyperbolic equivalence for quadric bundles and quadratic forms on vector bundles and show that hyperbolic equivalent quadric bundles share many important properties: they have the same Brauer data; moreover, if they have the same dimension over the base, they are birational over the base and have equal classes in the Grothendieck ring of varieties. Furthermore, when the base is a projective space we show that two quadratic forms are hyperbolic equivalent if and only if their cokernel sheaves are isomorphic up to twist, their fibers over a fixed point of the base are Witt equivalent, and, in some cases, certain quadratic forms on intermediate cohomology groups of the underlying vector bundles are Witt equivalent. For this we show that any quadratic form over $\mathbb{P}^n$ is hyperbolic equivalent to a quadratic form whose underlying vector bundle has many cohomology vanishings; this class of bundles, called VLC bundles in the paper, is interesting by itself.

The functor KR(X), defined in [4], is a contravariant functor defined in the category of spaces with involutions.It is shown herein that this functor is classified by equivariant maps into the complex Grassmann manifold, which is given the involution induced by complex conjugation.For the case of free involutions it is shown that the classifying maps can be taken to lie outside the fixed point set of the Grassmann manifold.This fixed point set can be identified with the real Grassmann manifold.It is then shown that, for free involutions, KR(X) is an invariant of the homotopy type of the orbit space X modulo its involution.The multiplicative group of real line bundles, real in the sense of [4], is shown to be classified by equivariant maps into a quadric surface Q in complex projective space.Q carries a free involution and this classification is again valid for spaces with free involutions. I. Introduction.In [4] the definition of Atiyah-real vector bundles was given, and they, together with the associated A-theory, were studied.The purpose of this paper is twofold.First we will extend the known classification theorems for complex A-theory to the new theory.Second we will obtain new results for the case of Atiyah-real vector bundles over spaces with free involutions.It was anticipated that this functor could be used to obtain a characterization of Mk, the differentiable seven sphere defined by Milnor in [1].This question is studied in 111, where KR{Ml) is computed.To review the definitions and results of [4], we have that a real space is a topological space X together with a homeomorphism t: X -> X with t2 = identity.We will write ?(x) = x.An Atiyah-real vector bundle over a real space is a complex vector bundle E -> X which itself is a real space and such that (1) 7r: 7i-> Xis equivariant, (2) the induced map Ex -> Es is conjugate linear.The Grothendieck group of Atiyah-real vector bundles over the real space X is denoted by KR(X).There is a Bott periodicity theorem and derived cohomology theory.Define(1) Rp-" = R'i + iRp,

The stable classification of quadratic forms over a field is given by the Witt group. Topology, via surgery theory, has embedded the Witt groups in a general theory of forms over any ring with involution. In this paper we use geometrically inspired methods to make computations. General results on the L-theory of a Laurent polynomial extension are used to study the Witt groups of genus 0 function fields.

A non-Hermitian operator may serve as the Hamiltonian for a unitary quantum system, if we can modify the Hilbert space of state vectors of the system so that it turns into a Hermitian operator. If this operator is time-dependent, the modified Hilbert space is generally time-dependent. This in turn leads to a generic conflict between the condition that the Hamiltonian is an observable of the system and that it generates a unitary time-evolution via the standard Schr\"odinger equation. We propose a geometric framework for addressing this problem. In particular we show that the Hamiltonian operator consists of a geometric part, which is determined by a metric-compatible connection on an underlying Hermitian vector bundle, and a non-geometric part which we identify with the energy observable. The same quantum system can be locally described using a time-dependent Hamiltonian that acts in a time-independent state space and is the sum of a geometric part and the energy operator. The full global description of the system is achieved within the framework of a moderate geometric extension of quantum mechanics where the role of the Hilbert space of state vectors is played by a Hermitian vector bundle $\mathcal{E}$ endowed with a metric compatible connection, and observables are given by global sections of a real vector bundle that is determined by $\mathcal{E}$. We examine the utility of our proposal to describe a class of two-level systems where $\mathcal{E}$ is a Hermitian vector bundle over a two-dimensional sphere.

In [2] Albert proved that a finite-dimensional absolute-valued algebra over the reals is necessarily aiternative (and hence the reals, complexes, quaternions, or Cayley numbers). In [3 ] he extended this from finite-dimensional to algebraic algebras. Recently, Wright [9] succeeded in removing the assumption that the algebra is algebraic. Wright proceeds by proving that the norm springs from an inner product, and then that the algebra is algebraic. Now if the norm I xI comes from an inner product, then xI 2 is a quadratic form in x, and moreover the assumption xy = x I y| means that it is a quadratic form admitting composition. Thus Albert's finite-dimensional theorem can be proved by combining Wright's result with Hurwitz's classical theorem on quadratic forms admitting composition. The main purpose of this paper is to make a similar method possible in the infinite-dimensional case by providing a suitable generalization of Hurwitz's theorem. The result is essentially that infinite-dimensional quadratic forms cannot admit composition, except in the rather trivial case of purely inseparable fields of characteristic two. In the concluding moments of the proof we rely heavily on the recent developments in the theory of alternative rings. Until then the paper is quite self-contained and elementary, and the style of the argument is very close to that of [1], [5], and [8]. Let A be a vector space over a field F. The function g(x) from A to F is a quadratic form if

The author describes a canonical structure on a Grassmannian fibration whose fiber is a Grassmann manifold of the tangent spaces of a smooth manifold. This structure generalizes the symplectic structure on the cotangent bundle. This symplectic form takes its values in a vector space or even in a vector bundle. This structure is canonical; it is uniquely defined by a smooth manifold. Bibliography: 5 titles.

We propose a definition of persistent Stiefel–Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual Čech filtration of such a subset can be endowed with a vector bundle structure, that we call a Čech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its first persistent Stiefel–Whitney class. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot (1,2). We illustrate our method on several datasets inspired by image analysis.

A result of André Weil allows one to describe rank$n$vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set$\text{GL}_{n}(\mathbb{A})$of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to$G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles$\mathbb{A}_{X}^{\bullet }$, we have an equivalence$\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$between perfect complexes on$X$and cartesian perfect complexes for$\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal$\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

Let X be a compact Riemann surface of genus g ≥ 3 . Let L = ( L , [ . , . ] , ♯ ) be a holomorphic Lie algebroid over X of rank one and degree ( L ) < 2 − 2 g . We consider the moduli space of holomorphic L λ -connections over X, where λ ∈ C . We compute the Picard group of the moduli space of L λ -connections by constructing an explicit smooth compactification of the moduli space of those L λ -connections whose underlying vector bundle is stable such that the complement is a smooth divisor. We also show that the automorphism group of the moduli space of L λ -connections fits into a short exact sequence that involves the automorphism group of the moduli space of stable vector bundle over X. For λ = 1, we get Lie algebroid de Rham moduli space of L -connections and we determine its Chow group. Communicated by Manuel Reyes

Given a quadratic form on a vector space, the geometric algebra of the corresponding pseudo-euclidean space is defined in terms of a simple set of rules which characterizes the geometric product of vectors. We develop geometric algebra in such a way that it augments, but remains fully compatible with, the more traditional tools of matrix algebra. Indeed, matrix multiplication arises naturally from the geometric multiplication of vectors by introducing a spectral basis of mutually annihiliating idempotents in the geometric algebra. With the help of a few more algebraic identities, and given the proper geometric interpretation, the geometric algebra can be applied to the study of affine, projective, conformal and other geometries. The advantage of geometric algebra is that it provides a single algebraic framework with a comprehensive, but flexible, geometric interpretation. For example, the affine plane of rays is obtained from the euclidean plane of points by adding a single anti-commuting vector to the underlying vector space. The key to the study of noneuclidean geometries is the definition of the operations of meet and join, in terms of which incidence relationships are expressed. The horosphere provides a homogeneous model of euclidean space, and is obtained by adding a second anti-commuting vector to the underlying vector space of the affine plane. Linear orthogonal transformations on the higher dimensional vector space correspond to conformal or Mobius transformations on the horosphere. The horosphere was first constructed by F.A. Wachter (1792–1817), but has only recently attracted attention byo ering a host of new computational tools in projective and hyperbolic geometries when formulated in terms of geometric algebra.

Given a symmetric bilinear form F on a finite dimensional real vector space V we attempt to find ‘nice’ bases where the matrix representation of the form is as simple as possible, in the same sort of way as we did in the Gram-Schmidt orthogonalization process for inner products. The theorem which says that this is possible and which describes all the possible matrix representations one can get is called Sylvester’s law of inertia. (Sylvester’s law of inertia has nothing to do with inertia: the story goes that Sylvester said, ‘If Isaac Newton can have a law of inertia, then so can I’, and consequently gave this theorem the name it has been known by ever since.) This ‘law’ is exactly analogous to Theorem 2.33 which classified all finite dimensional vector spaces using dimension; it turns out that symmetric bilinear forms are classified by the dimension of the underlying vector space and two further numbers, called the rank and the signature of the form.

We determine the Chow ring (with Q-coecients) of M6 by showing that all Chow classes are tautological. (In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective.) We stratify the moduli space into locally closed strata which are group quotients, and use a theorem of Vistoli to show that their Chow rings are generated by Chern classes of natural vector bundles. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus six case relies in addition on the particularly beautiful Brill{Noether theory in this case, and in particular on a rank ve vector bundle \relativizing a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the rank six vector bundle of quadrics cutting out the canonical curve.

The Kobayashi indicatrix (infinitesimal unit ball) of a domain in IE n is known to be a biholomorphic invariant. In particular, if a domain is biholomorphic to a ball, then the indicatrix is the ball. Until the recent deep results of Lempert [4], it was not known to what extent the indicatrix characterizes the domain. Sibony had shown earlier that the indicatrix of any pseudoconvex circular domain is the domain itself [10]; hence the indicatrix determines such domains up to biholomorphism. This is not true in general. Lempert has shown that, for smoothly bounded strictly convex domains, three invariants determine the domain up to biholomorphism and a linear change of variables at a base point. These invariants consist of not only the Kobayashi indicatrix but also a quadratic form and a hermitian form on a certain vector bundle. In two dimensions, there is an open set in a Frechet space, corresponding to domains not biholomorphic to the ball but with indicatrix the ball. The result of this paper is that, for ellipsoids, if the Kobayashi indicatrix is the ball then the ellipsoid is biholomorphic to the ball. This is proved in the following equivalent form: if an ellipsoid symmetric about 0 is not biholomorphic to the ball, then its infinitesimal Kobayashi metric at 0 is not hermitian. In this language, our result answers a question posed by Reiffen in [7] about the differential-geometric nature of the Carath6odory metric for ellipsoids. By two results of Lempert ([3] and [2]), for strictly convex domains the Carathe6dory and Kobayashi metrics coincide and determine a Finsler metric. One now sees that, for ellipsoids, the metric is not hermitian and hence, by a theorem of Reiffen [7], not Riemannian. We prove that the infinitesimal Kobayashi metric at 0 is not hermitian by proving that the parallelogram law fails to hold. An ellipsoid ~ may be written (via a linear change of coordinates) as )-], ] zj] 2 ..~ ,~j Re (z 2) < , 0 < Aj < 1. By a

The computation of odd dimensional projective surgery groups of finite groups