Stratified-algebraic vector bundles of small rank

We investigate vector bundles on real algebraic varieties. Our goal is to construct rank 2 real and complex stratified-algebraic vector bundles with prescribed Stiefel–Whitney and Chern classes, respectively. We obtain a partial solution of this problem and present two applications.

Problem 1.2. Given a positive integer k, characterize the cohomology classes u i in H 2i (X; Z) for which there exists a rank k stratified-algebraic C-vector bundle ξ on X with c i (ξ) = u i for 1 ≤ i ≤ k.
In [14] we solved Problems 1.1 and 1.2 for k = 1. Presently, we address these problems for k = 2 (Theorems 2.3 and 2.6). The case k ≥ 3 seems to be out of reach. Our partial solution of Problem 1.2 leads to a criterion for a topological C-vector bundle of rank 2 and a topological H-line bundle to admit a stratified-algebraic structure (Theorems 2.7 and 2.8).
It would be interesting to obtain at least a partial solution of the counterparts of Problems 1.1 and 1.2 for k = 2, where stratified-algebraic vector bundles are replaced by algebraic vector bundles. However, this would require some new approach, different from the methods used here.
Let X be a smooth manifold, and let M be a smooth codimension k submanifold of X. Suppose that the normal bundle to M in X is oriented, and denote by τ X M the Thom class of M in the cohomology group H k (X, X \ M ; Z), cf. [16, p. 118]. The image of τ X M by the restriction homomorphism H k (X, X \ M ; Z) → H k (X; Z), induced by the inclusion map X → (X, X \ M ), will be denoted by [[M ]] X and called the cohomology class represented by M . If X is compact and oriented, and M is endowed with the compatible orientation, then [[M ]] X is up to sign Poincaré dual to the homology class in H * (X; Z) represented by M , cf. [16, p. 136]. Similarly, without any orientability assumption, we define the cohomology class [M ] X in H k (X; Z/2) represented by M . The cohomology class [M ] X is Poincaré dual to the homology class in H * (X; Z/2) represented by M .
Let Y be a smooth manifold, and let N be a smooth submanifold of Y . Let f : X → Y be a smooth map transverse to N . If the normal bundle to N in Y is oriented and the normal bundle to the smooth submanifold M : [5,p. 117,Theorem 6.7]). In particular, Without any orientability assumption, Let ξ be a rank k smooth R-vector bundle on X. A smooth section s : X → ξ is said to be transverse regular if it is transverse to the zero section of ξ. In that case, the zero locus of s, is a smooth codimension k submanifold of X. We identify the normal bundle to Z(s) in X with ξ| Z(s) via the isomorphism induced by s. In particular, if the vector bundle ξ is oriented, then so is the normal bundle to Z(s) in X and where e(ξ) stands for the Euler class of ξ. Indeed, let E be the total space of ξ and p : E → X the bundle projection. Identify X with the image of the zero section of ξ. The section s is transverse to X and Z(s) = s −1 (X).
Recall that on a smooth manifold each topological vector bundle is isomorphic to a smooth vector bundle, which is uniquely determined up to smooth isomorphism, cf. [5, p. 101].
For any rank k F-vector bundle η, where F = R or F = C, let det η denote the kth exterior power of η. Thus det η is an F-line bundle. Furthermore, Given a subset A of X and a cohomology class u in H i (X; G), where G = Z/2 or G = Z, we denote by u| A the image of u by the homomorphism Proposition 2.1. Let X be a smooth manifold and let θ be a rank 2 topological R-vector bundle on X. Then there exist smooth submanifolds M i of X such that codim X M i = i, where ν is the normal bundle to M 2 in X, and Proof. We may assume without loss of generality that the vector bundle θ is smooth. Let Furthermore, since we identify ν with θ| M2 , we obtain det ν = (det θ)| M2 and as required. Proposition 2.1 provides motivation for the following result.

Proposition 2.2. Let X be a smooth manifold and let
where ν is the normal bundle to M 2 in X. Then there exists a rank 2 smooth R-vector bundle θ on X with Furthermore, the vector bundle θ can be chosen so that there exist smooth transverse regular sections Vol. 107 (2016)

Stratified-algebraic vector bundles 243
Proof. Let λ be a smooth R-line bundle on X with It is well known that there exists a smooth transverse regular section u : We assert that the R-line bundles μ and λ| T \M2 are isomorphic. Indeed, we have Consequently, (4) On the other hand, in view of (3), By combining (4) and (5), we get which implies the assertion, cf. [7, p. 234]. Let ε be the standard trivial R-line bundle on X with total space X × R and let τ : X → λ ⊕ ε be the smooth section defined by τ (x) = (0, (x, 1)) for all x in X. By the assertion, there exists a smooth isomorphism Let θ be the smooth R-vector bundle on X obtained by gluing ρ * ν and (λ ⊕ ε)| X\M2 over T \ M 2 using ϕ. Similarly, let s 2 : X → θ be the smooth section obtained by gluing σ and τ | X\M2 over T \M 2 using ϕ. By construction, θ is a rank 2 smooth R-vector bundle on X, the section s 2 is transverse regular, and In view of (1), (2), (6), (7), it remains to prove that the R-line bundles det θ and λ are isomorphic. To this end it suffices to show the equality cf. [7, p. 234]. Note that and hence As a portion of the long exact cohomology sequence of the pair (X, X \ M 2 ), we get where e : X \ M 2 → X is the inclusion map. Since H 1 (X, X \ M 2 ; Z/2) = 0 (cf. [16, pp. 106, 117], it follows that e * is a monomorphism. Hence (8) holds, as required.
If X is a smooth manifold, by combining Propositions 2.1 and 2.2, we obtain a characterization of the cohomology classes u i in H i (X; Z/2) for which there exists a rank 2 topological R-vector bundle θ on X with w i (θ) = u i for i = 1, 2. Our partial solution of Problem 1.1 is of a similar nature. We first recall a well-known phenomenon specific to real algebraic geometry. Namely, if X is a nonsingular real algebraic variety, it can happen that a nonsingular Zariski locally closed subvariety of X is Euclidean closed but not Zariski closed. Let K be a subfield of F, where K (as F) stands for R, C, or H. Any F-vector bundle η can be regarded as a K-vector bundle, which is indicated by η K . In Suppose now that ξ is a rank k smooth C-vector bundle on a smooth manifold X. Recall that where ξ R is endowed with the orientation induced by the complex structure, cf. [16, p. 158]. If s : X → ξ is a smooth transverse regular section, then we regard the normal bundle to Z(s) in X as a C-vector bundle, identifying it with ξ| Z(s) . In particular, the normal bundle to Z(s) in X is canonically oriented, the cohomology class [[Z(s)]] X in H 2k (X; Z) is defined, and If M is a smooth codimension 2k submanifold of X and the normal bundle ν to M in X is endowed with a complex structure, then the R-vector bundle Proof. We may assume without loss of generality that the vector bundle θ is smooth. Let Furthermore, since we identify ν 2 with θ| M2 , we obtain det ν 2 = (det θ)| M2 and as required.
The following is a counterpart of Proposition 2.2 for C-vector bundles.
Proposition 2.5. Let X be a smooth manifold, and let M i be a smooth submanifold of X such that codim X M i = 2i, the normal bundle ν i to M i in X is endowed with a complex structure for i = 1, 2, and Then there exists a rank 2 smooth C-vector bundle θ on X with Furthermore, the vector bundle θ can be chosen so that there exist smooth transverse regular sections Proof. The argument is analogous to that in the proof of Proposition 2.2. Let λ be a smooth C-line bundle on X with By [14,Lemma 8.20], there exists a smooth transverse regular section u : X → λ satisfying Z(u) = M 1 .
(2) Let ρ : T → M 2 be a tubular neighborhood of M 2 in X. There exists a smooth transverse regular section σ : T → ρ * ν 2 such that Z(σ) = M 2 . In particular, where ε σ is the trivial C-line subbundle of (ρ * ν 2 )| T \M2 generated by σ| T \M2 , and μ is a smooth C-line bundle on T \ M 2 . We assert that the C-line bundles μ and λ T \M2 are isomorphic. Indeed, we have Consequently, the map ρ : T → M 2 being a homotopy inverse of the inclusion map M 2 → T . Hence On the other hand, in view of (3), By combining (4) and (5), we get which implies the assertion, cf. [7, p. 234].
Let ε be the standard trivial C-line bundle on X with total space X × C, and let τ : X → λ ⊕ ε be the smooth section defined by τ (x) = (0, (x, 1)) for all x in X. By the assertion, there exists a smooth isomorphism Let θ be the smooth C-vector bundle on X obtained by gluing ρ * ν 2 and (λ ⊕ ε)| X\M2 over T \ M 2 using ϕ. Similarly, let s 2 : X → θ be the smooth section obtained by gluing σ and τ | X\M2 over T \M 2 using ϕ. By construction, θ is a rank 2 smooth C-vector bundle on X, the section s 2 is transverse regular, and Z(s 2 ) = M 2 .
In view of (1), (2), (6), (7), it remains to prove that the C-line bundles det θ and λ are isomorphic. To this end, it suffices to show the equality cf. [7, p. 234]. Note that and hence As a portion of the long exact cohomology sequence of the pair (X, X \ M 2 ), we get where e : X \ M 2 → X is the inclusion map. Since H 2 (X, X \ M 2 ; Z) = 0 (cf. [16, pp. 110, 117]), it follows that e * is a monomorphism. Hence (8) holds, as required.
If X is a smooth manifold, then Propositions 2.4 and 2.5 yield a characterization of the cohomology classes u i on H 2i (X; Z) for which there exists a rank 2 topological C-vector bundle θ on X with c i (θ) = u i for i = 1, 2.
Our partial solution of Problem 1.2 is the following.
Theorem 2.6. Let X be a compact nonsingular real algebraic variety, and let M i be a smooth submanifold of X such that codim X M i = 2i, the normal bundle ν i to M i in X is endowed with a complex structure for i = 1, 2, and Assume that M i is a nonsingular Zariski locally closed subvariety of X for i = 1, 2. Then there exists a rank 2 stratified-algebraic C-vector bundle ξ on X with Proof. It suffices to make use of Proposition 2.5 and [14, Theorem 1.9].
We conclude this paper by giving two applications of Theorem 2.6.
Theorem 2.7. Let X be a compact nonsingular real algebraic variety, and let θ be a rank 2 topological C-vector bundle on X. Let M i be smooth submanifolds of X such that codim X M i = 2i, the normal bundle ν i to M i in X is endowed with a complex structure, Proof. According to Theorem 2.6, there exists a rank 2 stratified-algebraic C-vector bundle ξ on X with c i (ξ) = c i (θ) for i = 1, 2. Consequently, Hence, if the condition on the torsion in the cohomology groups H 2k (X; Z) is satisfied, then the C-vector bundles ξ and θ are stably equivalent, cf. [17,Theorem 3.2]. This implies, in view of [14,Corollary 3.14], that θ admits a stratified-algebraic structure.
The second application concerns H-line bundles. integer k ≥ 3 the only torsion in the cohomology group H 2k (X; Z) is relatively prime to (k − 1)!, then λ admits a stratified-algebraic structure.
Proof. Suppose that the condition on the torsion in the cohomology groups H 2k (X; Z) is satisfied. By Theorem 2.7, the C-vector bundle λ C admits a stratified-algebraic structure. Hence, in view of [14,Theorem 1.7], the H-line bundle λ admits a stratified-algebraic structure.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.