Exciting news! Jie Liu has proven a conjecture of mine -- thereby resolving an old conjecture of Sommese. I'll briefly explain his result in this post.
The conjecture was:
Conjecture (L---, now proven by Jie Liu). Let X be a smooth projective variety over C, and E an ample vector bundle on X. Suppose that Hom(E,TX)≠0.
This conjecture was known (due to Andreatta-Wisniewski) in the case that there was a map E→TX of constant rank; Liu in fact proves something slightly stronger:
Theorem (Liu). Let X be a smooth projective variety over C, and suppose TX contains an ample subsheaf F. Then X≃Pn for some n.
The proof is quite nice. Old results of Araujo, Druel, and Kovacs immediately show that there exists an open subscheme X0⊂X and a map X0→T whose fibers are isomorphic to Pr for some r. We wish to show that T is a point. (Up to this point, this more or less follows Andreatta-Wisniewski). This is not particularly hard if the Pr-bundle X0→T is defined away from codimension two; indeed, in this case, one may find a proper curve in T and obtain a contradiction using standard techniques originally due to Peternell-Campana in this case -- this is essentially how Andreatta-Wisniewski proceed. The key fact is that if f:Y→C is a Pr-bundle over a proper curve C, the relative tangent sheaf Tf contains no ample subsheaves.
Unfortunately if F is not locally free, I don't see how to extend the fibration to codimension two, so this approach seems to be sunk.
Liu cleverly gets around this issue by working on an open subset of X rather than a closed subset. Unfortunately the paper is quite densely written and I do not as yet understand all the details -- if someone would like to explain what is happening "morally," I would love to hear it.
In any case, the results of the paper in which I made this conjecture allow one to deduce from this result an old conjecture of Sommese, which classifies smooth varieties containing a projective bundle as an ample divisor. This conjecture has attracted a lot of interest over the last few decades, so it's very gratifying to see it put to rest.