Hartshorne's famous conjecture on vector bundles say that any rank $2$ vector bundle over a projective space $\mathbb{P}^n$ with $n\geq 7$ splits into the direct sum of two line bundles.


So my questions are the following:


1) what is an evidence for this conjecture?


2)why is the condition on $n\geq 7$, but not other numbers?


3)any recent survey or reference on this conjecture? 



Evidences on Hartshorne's conjecture? References?
This [answer](https://mathoverflow.net/questions/12688/nonsingular-normal-schemes/12689#12689) of mine briefly discusses Hartshorne conjecture and some related questions about smooth subvarieties of $\mathbb P^n$ of small codimensions. It links to Hartshorne's original paper, which I think is still the best source to answer your questions 1) and 2).


As for 3), you can also look at Zolbani's [thesis](https://edocs.uis.edu/mmaji2/www/Research/Dissertation.pdf), which has a lot more details then his research statements mentioned by Steven. 


(That's all I know, I would be very interested in what's new about Hartshorne's conjecture as well).


**EDIT**: Today while answering another question I was reminded of a line of research which can be viewed as evidence for Hartshorne's conjecture: smooth subvarieties of small codimension behave cohomologically like complete intersections (this was discussed in Section 2 of Hartshorne original [paper](http://www.ams.org/bull/1974-80-06/S0002-9904-1974-13612-8/)). A paper by [Lyubeznik](http://www.jstor.org/pss/2946619), especially Section 11, has many such results, even for positive characteristic cases. It also includes many relevant references. 


