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I'm taking a first course on complex manifolds, and am trying to square what I hear with what I know of (real) differential geometry. Please forgive me if this question is misguided!

Here are two examples of ways of making vector bundles from codimension-one submanifolds.

  • (The tensor powers of its associated line bundle) In a complex manifold $M$ with a codimension-1 complex submanifold $D$, take as an atlas a system $(U_\alpha)$ of slice-coordinate charts for $V$, together with other charts $(V_\beta)$ covering $M\setminus D$. For each $n\in \mathbb{Z}$, define a line bundle via the following transition functions: $\phi_{\beta_1\beta_2}:V_{\beta_1}\cap V_{\beta_2}\to GL_1(\mathbb{C})$ is uniformly =1; $\phi_{\alpha_1\beta_2}:U_{\alpha_1}\cap V_{\beta_2}\to GL_1(\mathbb{C})$ is $z_1^n$, and $\phi_{\alpha_1\alpha_2}:U_{\alpha_1}\cap U_{\alpha_2}\to GL_1(\mathbb{C})$ is $z_1^n/w_1^n$, where $z_1$ and $w_1$ are the coordinates whose vanishing determines $D$ on $U_{\alpha_1}$ and $U_{\alpha_2}$ respectively.

Comment: This also seems to work fine if we replace "complex" by "smooth (real)" throughout. However, the family of line bundles isn't so interesting: the even ones are all trivial; the odd ones are mutually isomorphic.

  • (Vector bundles on spheres) For each homotopy class of maps $S^{n-1}\to GL_k(\mathbb{R})$, we can construct a vector bundle of rank $k$ on $S^n$, by using a representative of this class to define a transition function on the intersection of the "north" and "south" stereographic projection charts (which has $S^{n-1}$ as a retract).

I'd like to know: are these indeed analogous? Are they special cases of, say, a general method for constructing a smooth (respectively, complex) rank-$k$ vector bundle on a smooth (resp., complex) manifold out of a map from a codimension-one submanifold into $GL_k(\mathbb{R})$ (resp., $GL_k(\mathbb{C})$?

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Have you looked at the clutching construction in books on K-theory? – Tim Porter Feb 9 '10 at 16:39

I don't think these two constructions are particularly analogous.

The first construction works for any codimension 1 submanifold on any complex manifold, but only gives one line bundle (and its tensor powers). It's based on the idea that the submanifold should be the vanishing locus of a section of some line bundle.

The second construction is based on the idea that if a codimension-1 submanifold splits the manifold into two pieces, then we construct vector bundles on the entire space by gluing together vector bundles on the pieces. On the sphere, the two pieces are contractible, so the only interesting information is on the submanifold, and we can get every vector bundle on the sphere just from that data. In general, to construct a vector bundle, you would have to choose bundles on the pieces and a gluing function on the submanifold.

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