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What are some examples of algebraic varieties over the complex numbers with no (algebraic) vector bundles other than the trivial ones?

The only example I can think of is $ \mathbb{A}^n_{\mathbb{C}} $ and this is a very deep example - see the Quillen-Suslin theorem.

The inspiration for this question is the following fact that I learnt quite late in life: The 3-sphere $ S^3 $ does not have any topological real vector bundles other than the trivial ones. In fact, the excellent answer to this question tells us much more.

Cranium Clamp
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  • Any smooth varieties carries the tangent vector bundle, so you assumption implies that the tangent bundle is trivial. – Sasha May 17 '22 at 10:30
  • For projective varieties,(not a finite set of points) take an ample line bundle. Then since $h^{0}(L)>1$ it is non-trivial. – Nick L May 17 '22 at 12:14
  • @NickL you mean very* ample line bundle. Yes, no projective variety of dimension > 0 can be an example. – Cranium Clamp May 17 '22 at 16:40
  • I guess $\mathbb{P}^1$ comes closest in the projective case since all vector bundles are direct sums of $\mathcal{O}(n)$'s. -- and I think this is the only smooth variety with this property. You can come up with singular examples by chaining $\mathbb{P}^1$'s together, so maybe gluing some $\mathbb{A}^1$'s might be a place to start? – Ashwath Rabindranath May 18 '22 at 23:03

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