Separable rational connectedness and stability

In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability of the tangent sheaf (if there are no other obvious reasons). A simple application of the results proves that a smooth Fano complete intersection is separably rationally connected if and only if it is separably uniruled. In particular, a general such Fano complete intersection is separably rationally connected.Comment: 6 pages. Reference and acknowledgement added. Comments are welcom

Towards the symplectic Graber-Harris-Starr theorems

A theorem of Graber, Harris, and Starr states that a rationally connected fibration over a curve has a section. We study an analogous question in symplectic geometry. Namely, given a rationally connected fibration over a curve, can one find a section which gives a non-zero Gromov-Witten invariant? We observe that for any fibration, the existence of a section which gives a non-zero Gromov-Witten invariant only depends on the generic fiber, i.e. a variety defined over the function field of a curve. Some examples of rationally connected fibrations with this property are given, including all rational surface fibrations. We also prove some results, which says that in certain cases we can "lift" Gromov-Witten invariants of the base to the total space of a rationally connected fibration.Comment: 16 pages. Comments are welcom

Symplectic geometry of rationally connected threefolds

We study symplectic geometry of rationally connected $3$-folds. The first result shows that rationally connectedness is a symplectic deformation invariant in dimension $3$. If a rationally connected $3$-fold $X$ is Fano or $b_2(X)=2$, we prove that it is symplectic rationally connected, i.e. there is a non-zero Gromov-Witten invariant with two insertions being the class of a point. Finally we prove that many rationally connected $3$-folds are birational to a symplectic rationally connected variety.Comment: 25 pages. Preliminary version. Comments are welcom