Operators and Analytic Vector Bundles in H-Spaces

Let X be a compact complex analytic space with structure sheaf (9 and let Hi(X, GL(r, (9)) denote the cohomology set parameterizing isomorphism classes of rank r analytic vector bundles on X. If r > 1, Ha(X, GL(r, (9)) may not naturally form an analytic space; in fact, the natural topology on the cohomology set is not generally Hausdorff. In this paper we provide computable criteria for determining the topological non-separation of points in the cohomology set. For bundles ~o and ~p admitting a nonzero homomorphism h :~0~p, we show that under suitable hypotheses, it is possible to simultaneously deform q~, ~/~ and h in analytic oneparameter families ~o(t), ~p(t) and h(t) such that, for t near zero, h(t) defines an isomorphism between ~o(t) and W(t) converging as t ~ 0 to the homomorphism h from ~o = ~0(0) to ~0 =~p(0). In case X is a Riemann surface, this method yields necessary and sufficient conditions for non-separation of rank two bundles.

It is well known that a compact analytic manifold carries, besides the usual topological invariants (its Euler characteristic, the Betti numbers), some numerical invariants like the degree of the canonical divisor, the Hodge numbers, the arithmetic genus, and so on. A common property of all these invariants is that they coincide with the dimension of the cohomology spaces of certain analytic vector bundles or analytic sheaves. In this way, these invariants become accessible to algebraic and homological methods.

On curvature and harmonic forms with values in analytic vector bundles.

SynopsisThis is the second of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears trie phenomenon of “crossing” of the exponents of singularities. In Part I, we introduced for the Dirichlet problem appropriate combinations of the simple tensor product singularities.In this second part, we extend the results of Part I to general non-homogeneous boundary conditions. Moreover, we show how these combinations of singularities appear in a natural way as sections of an analytic vector bundle above the edge. In the case when the interior operator is the Laplacian, we give a simpler expression of the combined singular functions, involving divided differences of powers of a complex variable describing the coordinates in the normal plane to the edge.

An important part of the classical theory of real or complex manifolds is the theory of (smooth, real analytic or complex analytic) vector bundles. With any vector bundle over a manifold (M,F) the sheaf of its (smooth, real analytic or complex analytic) sections is associated which is a locally free sheaf of F-modules, and in this way all the locally free sheaves of F-modules over (M,F) can be obtained. In the present paper, locally free sheaves of O-modules over a complex analytic supermanifold (M,O) are studied. The main results of the paper are the following ones. Given a locally free sheaf E of O-modules over a complex analytic supermanifold (M,O), we construct a locally free sheaf over the retract of (M,O) which is called the retract of E. Our first result is a classification of locally free sheaves of modules which have a given retract in terms of non-abelian 1-cohomology. The case of the tangent sheaf of a split supermanifold is studied in more details. Then we study locally free sheaves of modules over projective superspaces. A spectral sequence which connects the cohomology with values in a locally free sheaf of modules with the cohomology with values in its retract is constructed.

Analytic cycles and vector bundles on non-compact algebraic varieties

Let X be a real Banach space with an unconditional basis (for example, X = ℓ2 Hilbert space), let Ω ⊂ X be open, let M ⊂ Ω be a closed split real analytic Banach submanifold of Ω, let E → M be a real analytic Banach vector bundle, and let 𝒜 E → M be the sheaf of germs of real analytic sections of E → M. We show that the sheaf cohomology groups Hq(M, 𝒜 E) vanish for all q ⩾ 1, and there is a real analytic retraction r:U → M from an open set U with M ⊂ U ⊂ Ω such that r(x) = x for all x ∈ M. Some applications are also given, for example, we show that any infinite-dimensional real analytic Hilbert submanifold of separable affine or projective Hilbert space is real analytically parallelizable.

Let a differential field K with a derivation f↦f′ be given. A differential module over K has been defined as a K-vector space M of finite dimension together with a map ∂: M→M satisfying the rules: ∂(m1+m2)=∂(m1)+∂(m2), and ∂(fm)=f′m+f∂(m). In this definition, one refers to the chosen derivation of K. We want to introduce the more general concept of connection, which avoids this choice. The advantage is that one can perform constructions, especially for the Riemann-Hilbert problem, without reference to local parameters. To be more explicit, consider the field K=C(z) of the rational functions on the complex sphere P=C∪{∞}. The derivations that we have used are \( \frac{d}{{dt}} \) and tN \( {t^N}\frac{d}{{dt}} \) where t is a local parameter on the complex sphere (say t is z−a or 1/z or an even more complicated expression). The definition of connection (in its various forms) requires other concepts such as (universal) differentials, analytic and algebraic vector bundles, and local systems. We will introduce those concepts and discuss the properties that interest us here. KeywordsVector BundleLine BundleGlobal SectionDifferential ModuleSingular LocusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Throughout this paper X denotes a real non-singular affine algebraic variety of dimension n. We will give a realization of the characteristic classes (the Stiefel-Whitney classes, the Pontrjagin classes and the Euler classes) of real affine algebraic vector bundles over X by algebraic subvarieties (Theorems 1, 2). For the complex field, Grothendieck [4] showed that the Chern classes of an algebraic vector bundle over a complex non-singular quasi-projective variety are realized by algebraic cycles. Morimoto [6] considered the complex analytic case. We prove Theorems 1, 2 by the method used there. If we work over a real analytic vector bundle, Thorn's transversality theorem shows easily a realization of the characteristic classes by analytic subsets (see Suzuki [10]). Theorem 1 was partially proved in [2], [8], and two different applications of them were given in [2], [9]. In Section 4, Theorems 3, 4 will show that the smoothing of algebraic subvarieties of X of codimension 1 for homological equivalence is always possible. The proof uses an idea in [8]. Given two cohomology classes of X which are realized by algebraic subvarieties, it seems likely that their cup product is realized by an algebraic subvariety. We prove this under some assumptions, applying Theorems 1, 2 (Theorem 5). We must remark that a realization of the cup product by an analytic subset is always possible according to the transversality theorem. Section 6 considers an affine algebraic structure of a topological vector bundle over X. If the rank is 1, and if the Stiefel-Whitney class is realized by an algebraic subvariety, then the bundle has an affine algebraic structure.

Introduction Chapter 1: Preparatory material 1. Multiplicative sequences 2. Sheaves 3. Fibre bundles 4. Characteristic classes Chapter 2: The cobordism ring 5. Pontrjagin numbers 6. The ring /ss(/Omega) /oplus //Varrho 7. The cobordism ring /omega 8. The index of a 4k-dimensional manifold 9. The virtual index Chapter 3: The Todd genus 10. Definiton of the Todd genus 11. The virutal generalised Todd genus 12. The t-characteristic of a GL(q, C)-bundle 13. Split manifolds and splitting methods 14. Multiplicative properties of the Todd genus Chapter 4: The Riemann-Roch theorem for algebraic manifolds 15. Cohomology of Compact complex manifolds 16. Further properties of the (/chi)x characteristics 17. The virtual (/chi)x characteristics 18. Some fundamental theorems of Kodaira 19. The virtual (/chi)x characteristics for algebraic manifolds 20. The Riemann-Roch theorem for algebraic manifolds and complex analytic line bundles 21. The Riemann-Roch theorem for algebraic manifolds and complex analytic vector bundles Appendix 1 by R.L.E. Schwarzenberger 22. Applications of the Riemann-Roch theorem 23. The Riemann-Roch theorem of Grothendieck 24. The Grothendieck ring of continuous vector bundles 25. The Atijah-Singer index theorem 26. Integrality theorems for differentiable manifolds Appendix 2 by A. Borel A spectral sequence for complex analytic bundles Bibliography Index

Some properties of complex analytic vector bundles over compact complex homogeneous spaces

The theory of complex line bundles, which was developed by A.Weil and K. Kodaira, D

The structure of the section space of a real analytic vector bundle on a real analytic manifold $X$ is studied. This is used to improve a result of Grothendieck and Poly on the zero spaces of elliptic operators and to extend a result of Domański and the

We develop a version of Sen theory for equivariant vector bundles on the Fargues-Fontaine curve. We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle. A comparison with the theory of (ϕ, )-modules in the cyclotomic case then recovers the Cherbonnier-Colmez decompletion theorem. Next, we focus on the subcategory of de Rham locally analytic vector bundles. Using the p-adic monodromy theorem, we show that each locally analytic vector bundle E has a canonical differential equation for which the space of solutions has full rank. As a consequence, E and its sheaf of solutions Sol(E) are in a natural correspondence, which gives a geometric interpretation of a result of Berger on (ϕ, )-modules. In particular, if V is a de Rham Galois representation, its associated filtered (ϕ, N , G K )-module is realized as the space of global solutions to the differential equation. A key to our approach is a vanishing result for the higher locally analytic vectors of representations satisfying the Tate-Sen formalism, which is also of independent interest. 1. Introduction 899 2. Locally analytic and pro-analytic vectors 905 3. Equivariant vector bundles 909 4. Locally analytic vector bundles 911 5. Acyclicity of locally analytic vectors for semilinear representations 916 6. Descent to locally analytic vectors 931 7. The comparison with (ϕ, )-modules 935 8. Locally analytic vector bundles and p-adic differential equations 938 Acknowledgments 945 References 945 MSC2020: primary 11F80; secondary 11S25.

Branched affine and projective structures on compact Riemann surfaces