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Post Made Community Wiki by S. Carnahan
2 Added two colleagues' opinion on the status of Hartshorne's question.

In the seventies and eighties of the preceding century, existence and classification of vector bundles on projective space $\mathbb P^n$ were all the rage, with contributions from such luminaries as Artin, Atiyah, Hartshorne and Mumford among many others. I have the feeling that not much progress has been made since.

For example, as far as I know, Hartshorne's apparently naïve question "Does there exist an indecomposable algebraic vector bundle of rank 2 on $\mathbb P^n_k \; ?\:"$ is still open for all fields $k$ and all integers $n\geq 6$.

Update[Next day] My colleagues André Hirschowitz and Arnaud Beauville, who are well informed about these questions, have allowed me to report that they feel quite confident that Hartshorne's question is still unsolved.

1

In the seventies and eighties of the preceding century, existence and classification of vector bundles on projective space $\mathbb P^n$ were all the rage, with contributions from such luminaries as Artin, Atiyah, Hartshorne and Mumford among many others. I have the feeling that not much progress has been made since.

For example, as far as I know, Hartshorne's apparently naïve question "Does there exist an indecomposable algebraic vector bundle of rank 2 on $\mathbb P^n_k \; ?\:"$ is still open for all fields $k$ and all integers $n\geq 6$.