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Is there any classification of the rank 2 complex vector bundles over $\mathbb CP^2$ up to diffeomorphism?

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    $\begingroup$ ... I see a theme forming. $\endgroup$ Jul 21, 2016 at 11:24
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    $\begingroup$ You may want to look into Okenek-Schneider-Spindler "Vector bundles on complex projective spaces". There is a section there discussing this question. $\endgroup$
    – Sasha
    Jul 21, 2016 at 11:29
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    $\begingroup$ I'm curious as to why you're tagging these questions Algebraic Geometry. Are you really interested in algebraic vector bundles? $\endgroup$
    – Mark Grant
    Jul 21, 2016 at 13:23
  • $\begingroup$ I just want to find find more explicit expression of these vector bundles, I guess this could connect to some knowledge in algebraic geometry. Sorry. $\endgroup$
    – DLIN
    Jul 22, 2016 at 2:00

2 Answers 2

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The question was essentially answered by Sasha's comment, but I will give a couple of additional/alternative references/remarks.

As a first remark, the continuous and smooth classifications coincide, this is discussed e.g. in the answers to this MO-question: From Topological to Smooth and Holomorphic Vector Bundles

The continuous classification of rank 2 complex vector bundles on $\mathbb{CP}^2$ is given exactly in terms of Chern classes. This can essentially be proved by obstruction theory (compute homotopy groups of $BU(2)$, identify the lifting classes in $H^{2i}(X;\mathbb{Z})$ with the Chern classes). More information can be found in

  • F.P. Peterson. Some remarks on Chern classes. Ann. Math. 69 (1959), 414-420.

Theorem 3.1 of that paper shows that there is an abelian group structure on the set of isomorphism classes of rank 2 bundles on $\mathbb{CP}^2$. Theorem 3.2 shows that a rank $2$ vector bundle on $\mathbb{CP}^2$ is trivial if and only if its Chern classes are trivial. (The triviality here is in the topological or the smooth category.)

The realizability of Chern classes $c_1,c_2$ by rank 2 vector bundles can also be derived from the obstruction theory. There is also a classical reference constructing rank 2 vector bundles on algebraic surfaces with prescribed Chern classes:

  • R.L.E. Schwarzenberger. Vector bundles on algebraic surfaces. Proc. London Math. Soc 11 (1961), 601-622.

This gives a more explicit construction which alternatively is known under the name Hartshorne-Serre correspondence.

Finally, since the question is tagged algebraic geometry, let me point out that the algebraic or holomorphic classification of rank 2 bundles on $\mathbb{CP}^2$ is very complicated. After fixing Chern classes and an additional invariant called the splitting type, the resulting moduli space of bundles has infinitely many irreducible components (so that there are uncountably many algebraic structures on the trivial rank 2 bundle).

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As additional comment, We have Cartan-Serre theorem for construction of rank 2 vector bundles on projetive varieties.

Let $X$ be a complex manifold, $L_1, L_2 ∈ Pic(X)$ are line bundles, $Z ⊂ X$ with $codim_X(Z) = 2$. Under some cohomological conditions, the sheaf $E$ sitting in

$$0 \to L_1 \to E \to L_2 \otimes J_Z → o$$ is locally free. Moreover, If $X$ is projective, any rank-2 holomorphic vector bundle can be constructed this way

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    $\begingroup$ For any compact smooth complex surface $X$ and any pair of Chern classes $c_1,c_2,$ one defines the discriminant $Δ(2,c_1,c_2):=1/2(c_2−1/4c^2_1)$. When $X$ is non-algebraic, a necessary condition for the existence of a rank-2 holomorphic vector bundle with these Chern classes is $Δ(2,c_1,c_2)≥0;$ . See this paper M. Aprodu, V. Brînzănescu and M. Toma, Math. Z. 242 (2002), no. 1, 63–73 $\endgroup$
    – user21574
    Jul 29, 2017 at 13:17
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    $\begingroup$ There is complete classification of uniform rank-2 vector bundles on Fano manifolds under some assumptions. See Roberto Muñoz, Gianluca Occhetta, and Luis E. Solá Conde, On rank 2 vector bundles on Fano manifolds, Kyoto J. Math. Volume 54, Number 1 (2014), 167-197. $\endgroup$
    – user21574
    Jul 29, 2017 at 13:34

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