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Motivation: $T_{\mathbb P^2}$ isn't an extension of line bundles

Here's a trick to show that the tangent bundle $T$ of $\mathbb P^2$ is not an extension of line bundles. If it were, we would have a short exact sequence $$\def\O{\mathcal O} 0\to \O(a)\to T\to \O(b)\to 0 $$ for some integers $a$ and $b$. Then we can compute the total Chern class $$\begin{align*} c(T)& =c(\O(a))\cdot c(\O(b)) \\ &= (1+aH)(1+bH) \\ &= 1+(a+b)H+abH^2, \end{align*}$$

where $H=c_1(\O(1))$ is the class of a hyperplane.

On the other hand, we have the Euler sequence $$ 0\to \O\to \O(1)^3\to T\to 0 $$ which tells us that $$\begin{align*} c(T)&=c(T)\cdot c(\O)=c(\O(1)^3)\\ &=c(\O(1))^3= 1+3H+3H^2. \end{align*}$$

Now observe that there do not exist integers $a$ and $b$ so that $a+b=ab=3$, so $T$ cannot be an extension of line bundles.


The Question

More generally, whenever we have an extension of vector bundles $0\to L\to E\to M\to 0$, we have $c(E)=c(L)\cdot c(M)$. So to show that $E$ has no sub-bundles (or equivalently, no quotient bundles), it suffices to show that $c(E)$ doesn't factor. The question is whether the converse is true:

Suppose $E$ is a rank $r$ vector bundle on a (smooth quasi-projective) scheme (or manifold) $X$ so that $c(E)=c(L)c(M)$ for vector bundles $L$ and $M$ of rank $i$ and $r-i$, respctively. Must $E$ have a sub-bundle or rank $i$ or $r-i$?

Remark 1: The phrasing is a bit strange compared to the natural-sounding "If the total Chern class of a vector bundle factors, does it have a sub-bundle?" The point is that knowing the rank of $E$ is very important. We showed that $T_{\mathbb P^2}$ has no sub-bundles, but $O(1)^3$ has the same total Chern class and clearly has lots of sub-bundles.

Remark 2: Does either $L$ or $M$ have to be a sub-bundle of $E$? NO! For example, on $\mathbb P^1$, we have that $$ c(\O(1)\oplus \O(-1)) = (1+H)(1-H)=1 = c(\O)c(\O) $$ but $\O(1)\oplus \O(-1)$ doesn't have a sub-bundle isomorphic to $\O$ (because it has no non-vanishing sections).

Remark 3: What is the answer in the case $X=\mathbb P^n$?

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2 Answers 2

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The answer for projective spaces is negative. I think the simplest example are 2-bundles on $\mathbb{P}^3(\mathbb{C})$. In that case the Schwarzenberger condition is that $c_1c_2$ should be even. Atiyah and Rees have proved that for any pair $(c_1,c_2)$ satisfying this there are holomorphic vector bundles $\xi$ with $c_1(\xi)=c_1,c_2(\xi)=c_2$ (see Atiyah, Rees, Vector bundles on projective 3-space. Invent. Math. 35 (1976), 131–153.). The number of topologically distinct such bundles in 1 when $c_1$ is odd and 2 when $c_1$ is even. So e.g. there is a topologically nonsplit 2-bundle on $\mathbb{P}^3$ with total Chern class $(1+ka)(1-ka)$ where $a=c_1(\mathcal{O}(1))$.

The topological classification of 2-bundles on $\mathbb{P}^3$ and the existence of a holomorphic structure on them are also proved in Okonek, Schneider, Spindler, Vector bundles on complex projective spaces, chapter 1, 6.3.

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If you are also asking about the case of topological complex vector bundles over manifolds, consider the case $X=S^5$. There are no nontrivial rank $1$ bundles, but there is a nontrivial rank $2$ bundle, and of course its Chern class $1$ factors as $1\times 1$.

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