A projective variety $X$ is convex, if for any $f:\mathbb{P}^1 \to X$, the group $H^1(\mathbb{P}^1, f^\*(T\_X))$ vanishes. A big group of examples of convex varieties is made of homogeneous varieties. An homogeneous variety is the quotient variety $G/P$ of a Lie group $G$ by a parabolic subgroup $P$ of it.


My question is: is there an easy example of a projective variety, possibly smooth, that is convex but that does not admit such a description?


In the lectures: <http://www.math.uic.edu/~coskun/utah-notes.pdf>, the author asks whether every rationally connected smooth convex projective variety must also be homogeneous, thus suggesting that it should be rather easy to find such an example in the realm of non-rationally connected varieties.



Convex varieties that are not homogeneous
Of course every variety containing no rational curves is convex, by default. For a less trivial example, take the product of one such variety (for example, an abelian variety) with a homogeneous variety.


