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Was it proven already that smooth rational complex projecitve varieties are symplectically rationally connected? I.e. some GW invariant with two point insertions is non zero. What about smooth toric varieties?

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    $\begingroup$ That is an open conjecture. The major progress is by Zhiyu Tian, who proved the conjecture in dimension 3. "Symplectic geometry of rationally connected threefolds." Duke Math. J. 161, no. 5 (2012), pp. 803--843. $\endgroup$
    – Jason Starr
    Commented Aug 3, 2017 at 10:28
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    $\begingroup$ . . . I missed the word "rational" in the question. I doubt that there is much more known in the rational case than in the general rationally connected case. However, in the toric case, it is likely that more can be proved. $\endgroup$
    – Jason Starr
    Commented Aug 3, 2017 at 10:30
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    $\begingroup$ Thanks Jason! The abstract of this paper on arxiv seem to claim a bit less than what you say (I imaging the final version proves a stronger result) $\endgroup$
    – aglearner
    Commented Aug 3, 2017 at 11:34
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    $\begingroup$ Here is an idea in the toric case. This idea is close to the proof by Yifei Chen and Shokurov of "strong rational connectedness" for toric varieties. The "universal torsor" over a smooth, projective toric variety is an open subset of an affine space, and the toric variety is a GIT quotient of the universal torsor. For a general pair of points in that open subset, there is a unique line in the affine space connecting the points. The (rational) image is a rational curve in the toric variety. Is the corresponding $2$-point Gromov-Witten invariant of this curve class equal to $1$? $\endgroup$
    – Jason Starr
    Commented Aug 3, 2017 at 14:00
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    $\begingroup$ @JasonStarr There is no hope doing this using just the classes of the two points and no other classes. On $\mathbb P^1 \times \mathbb P^1$, the space of rational $(a,b)$ curves through two general points is $2a+2b-3$-dimensional if $a>0,b>0$ and is empty otherwise, so it is never zero-dimensional. I think your construction produces a $(1,1)$ curve and thus a one-dimensional moduli space. $\endgroup$
    – Will Sawin
    Commented Aug 3, 2017 at 14:58

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