## Duke Mathematical Journal

### Symplectic geometry of rationally connected threefolds

Zhiyu Tian

#### Abstract

We study the symplectic geometry of rationally connected $3$-folds. The first result shows that rational connectedness is a symplectic deformation invariant in dimension $3$. If a rationally connected $3$-fold $X$ is Fano or has Picard number $2$, we prove that there is a nonzero Gromov–Witten invariant with two insertions being the class of a point. That is, $X$ is symplectic rationally connected. Finally we prove that many rationally connected $3$-folds are birational to a symplectic rationally connected variety.

#### Article information

Source
Duke Math. J., Volume 161, Number 5 (2012), 803-843.

Dates
First available in Project Euclid: 27 March 2012

https://projecteuclid.org/euclid.dmj/1332866804

Digital Object Identifier
doi:10.1215/00127094-1548398

Mathematical Reviews number (MathSciNet)
MR2941881

Zentralblatt MATH identifier
1244.14041

#### Citation

Tian, Zhiyu. Symplectic geometry of rationally connected threefolds. Duke Math. J. 161 (2012), no. 5, 803--843. doi:10.1215/00127094-1548398. https://projecteuclid.org/euclid.dmj/1332866804

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