Timeline for Are rational varieties symplectically rationally connected?
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| Aug 3, 2017 at 15:02 | comment | added | Will Sawin | @JasonStarr I guess the problem here is that given two general points in the toric variety, there are multiple ways to lift them to the universal cover and thus multiple straight lines between them, which do not in general map to the same rational curve in the toric variety. | |
| Aug 3, 2017 at 14:58 | comment | added | Will Sawin | @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. | |
| Aug 3, 2017 at 14:00 | comment | added | Jason Starr | 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 $1$? | |
| Aug 3, 2017 at 12:26 | comment | added | Jason Starr | I did not read carefully enough your post. I agree that it is not yet known whether there is a nonzero, two-point, genus $0$ Gromov-Witten invariant. Tian did prove symplectic invariance of rational connectedness. | |
| Aug 3, 2017 at 11:34 | comment | added | aglearner | 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) | |
| Aug 3, 2017 at 11:31 | history | edited | aglearner |
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| Aug 3, 2017 at 10:30 | comment | added | Jason Starr | . . . 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. | |
| Aug 3, 2017 at 10:28 | comment | added | Jason Starr | 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. | |
| Aug 3, 2017 at 9:47 | history | asked | aglearner | CC BY-SA 3.0 |