Secondary characteristic classes and the Euler class

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,

Let X = Spec(A) be a smooth affine variety of dimension n ≥ 2 over a field k and P be a projective A-module of rank n. A result of Murthy ([Mu], Theorem 3.8) says that if k is an algebraically closed field and if the top Chern class Cn(P) in CH0(X) is zero, then P splits off a free summand of rank 1 (i.e. P A⊕ Q). We note that over any base field, the vanishing of the top Chern class is a necessary condition for P to split off a free summand of rank 1. However, as is shown by the example of the tangent bundle of an even dimensional real sphere, this condition is not sufficient. In view of these examples, it is of interest to explore whether one can classify examples of projective modules over smooth real varieties which have the property that the top Chern class of the projective module vanishes but the projective module does not split off a free summand of rank one. More precisely we ask the following:

Witten's top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with r-spin structures. It plays a key role in Witten's conjecture relating to the intersection theory on these moduli spaces. Our first goal is to compute the integral of Witten's class over the so-called double ramification cycles in genus 1. We obtain a simple closed formula for these integrals. This allows us, using the methods of [15], to find an algorithm for computing the intersection numbers of the Witten class with powers of the \psi-classes (or tautological classes) over any moduli space of r-spin structures, in short, all numbers involved in Witten's conjecture.

The usual expression for the secondary characteristic classes is regularized so that it fits for the case of Hilbert bundles. This gives index formulas for pseudodifferential operators with operator-valued symbols.

In this chapter, we introduce the relative group homology for a group-pair \(K \subset G\). The relative group homology is known to be a powerful method, e.g., for algebraic K-theory, secondary characteristic classes, and stability problems; see [Knu, Neu2, Dup, Zi1, AC]. For the purpose, in Sect. 7.1, we begin by reviewing the usual group homology. After that, we will study two versions of relative group homology. One introduces the relative group homology from the viewpoint of mapping cones, in Sect. 7.2. On the other hand, in Sect. 7.3, we discuss the Hochschild relative homology and a relation to the former version. After that, we give some examples, where the concept of malnormality is important. Furthermore, in Sect. 7.4, we explicitly give some cocycles of relative group cohomology. Throughout this chapter, we represent a group by G and a right \(\mathbb {Z}[G]\)-module by M.

We study a Hopf algebroid, \mathcal H , naturally associated to the groupoid U^δ_n ⋉ U_n . We show that classes in the Hopf cyclic cohomology of \mathcal H can be used to define secondary characteristic classes of trivialized flat U_n -bundles. For example, there is a cyclic class which corresponds to the universal transgressed Chern character and which gives rise to the continuous part of the ρ -invariant of Atiyah–Patodi–Singer. Moreover, these cyclic classes are shown to extend to pair with the K-theory of the associated C ^* -algebra. This point of view gives leads to homotopy invariance results for certain characteristic numbers. In particular, we define a subgroup of the cohomology of a group analogous to the Gelfand–Fuchs classes described by Connes [3] and show that the higher signatures associated to them are homotopy invariant.

We discuss the computation of the secondary characteristic classes and of the holonomy classes for the leaves of locally homogeneous subfoliations. Furthermore, we shall construct some examples of locally homogeneous subfoliations (F1, F2) so that a leaf (L1, L2) of (F1, F2) admits non-trivial secondary characteristic classes and non-trivial holonomy classes which cannot be obtained if we consider each leaf separately.

Algebraic Vector Bundles

In recent years there has been some interest in rank2 algebraic (or holomorphic) vector bundles over the complex projective space pn. One of the main reasons for this interest is the connection of such bundles with projective varieties of codimension 2 and in particular with the question whether such varieties are necessarily complete intersections. In [2] Hartshorne formulates the conjecture that every rank2 algebraic bundle over pn splits as the sum of 2 line bundles at least if n > 7. Rank 2 bundles over pa are classified topologically by their Chern classes cl, c 2. In [1] Atiyah and Rees carry out the topological classification for p3 and show the existence of "sufficiently many" rank 2 topological bundles over p4. On p2 and p3 every topological 2-plane bundle has an algebraic structure. It is not known whether this is true for n > 4. In [6] and [-7] non-trivial 2-plane bundles are constructed over P", n>5 , with c a = c 2 = 0 . These are possible candidates for topological bundles without algebraic structure. In this paper we carry the classification of topological 2-plane bundles over P~ a few steps further. Although this by no means solves the classification problem for algebraic bundles, we hope the results may be of some use to those interested in algebraic bundles and projective varieties. In the following we shall not distinguish between a map f:X~BU(2), its homotopy:class and the induced isomorphism class of rank 2 bundles. Thus we shall often write r/$X or q :X~BU(2) . As H*(P";2~) is a truncated polynomial algebra on a 2-dimensional generator x, the Chern classes cl, c 2 are integer multiples of x resp. x 2. Therefore we shall regard them as integers. Let us write

In this paper, by considering a bundle of algebras on a manifold, we construct a vector bundle that naturally contains the tangent bundle of that manifold and has the most important properties of tangent bundles. This vector bundle can be considered as an extended tangent bundle associated to that algebra bundle. To this end, we consider the algebra of the sections of that algebra bundle and find all derivations on this algebra. These derivations are related to the sections of a vector bundle that is the desired bundle. We also investigate the concepts of connection and metric on this extended tangent bundle and prove the existence of the Levi-Civita connection of a metric on the extended tangent bundle.

Introduction. In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider complex analytic connections in complex analytic fibre bundles. The situation is then radically different from that in the differentiable case. In the differentiable case connections always exist, but may not be integrable; in the complex analytic case connections may not exist at all. In both cases we are led therefore to certain obstructions, an obstruction to the integrability of a connection in the differentiable case, an obstruction to the existence of a connection in the complex analytic case. It is a basic theorem that, if the structure group is compact, the obstruction in the differentiable case (the curvature) generates the characteristic cohomology ring of the bundle (with real coefficients). What we shall show is that, in a large class of important cases, the obstruction in the complex analytic case also generates the characteristic cohomology ring. Using this fact we can then give a purely cohomological definition of the characteristic ring. This has a number of advantages over the differentiable approach: in the first place the definition is a canonical one, not depending on an arbitrary choice of connection; secondly we remain throughout in the complex analytic domain, our characteristic classes being expressed as elements of cohomology groups with coefficients in certain analytic sheaves; finally the procedure can be carried through without change for algebraic fibre bundles. The ideas outlined above are developed in considerable detail, and they are applied in particular to a problem first studied by Weil [17], namely the problem of characterizing those fibre bundles which arise from a representation of the fundamental group. We show how Weil's main result fits into the general picture, and we discuss various aspects of the problem. As no complete exposition of the theory of complex analytic fibre bundles has as yet been published, this paper should start with a basic exposition of this nature. However this would be a major undertaking in itself, and instead we shall simply summarize in ?1 the terminology and results on vector bundles which we require, and for the rest we refer to Grothendieck [8], Serre [12], and Hirzebruch [9].

On any smooth affine surface over an algebraically closed field, it is easy to construct rank two projective modules with given determinant as the first Chern class and an arbitrary rational equivalence class of a zero cycle as the second Chern class. We will briefly indicate this method in Section 1. Here we construct projective modules over smooth affine 3-folds with prescribed Chern classes and of the right rank. That is to say, we construct rank two projective modules with prescribed first and second Chem classes and rank three projective modules with prescribed first, second and third Chern classes (Theorem 2.1). As a corollary, using Roitman's theorem on torsion in zero cycles and Suslin's cancellation theorem, one gets that a rank three projective module over a smooth affine threefold with top Chem class zero has a free direct summand of rank one (Cor. 2.4). For example, over a rational threefold any projective module splits into a free module and a rank two projective module. Thus we obtain necessary and sufficient conditions in terms of appropriate Chern classes for modules to be efficiently generated. For example, on an affine rational 3-fold any line bundle is 3-generated and any rank two projective module is 4-generated. We also prove that any maximal ideal over a smooth affine variety of dimension d is the zero of a section of a rank d projective module (Theorem 3.1). For an affine 3-fold, if A3(X) = 0, we prove the validity of the Eisenbud-Evans estimate for finitely generated modules. We do not know the answers for most of these questions in higher dimensions. Sometimes we assume that the characteristic of the ground field is different from 2, 3 and 5 due to technical reasons.**

Bott periodicity and calculus of Euler classes on spheres

Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory Zachary Alexander Maddock This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anti-canonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semi-stable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry.

In differential topology it is often useful to be able to find restrictions on the possible vector bundles over a given manifold. For the non-equivariant case these restrictions usually state that some rational multinomial in the various charac teristic classes is an integral multiple of the fundamental cocyle.