Weighted blowup correspondence of orbifold Gromov--Witten invariants and applications

Abstract

Let $\mathsf{X}$ be a symplectic orbifold groupoid with $\mathsf{S}$ being a symplectic sub-orbifold groupoid, and ${\mathsf{X}}_{\mathfrak{a}}$ be the weight-$\mathfrak{a}$ blowup of $\mathsf{X}$ along $\mathsf{S}$ with $\mathsf{Z}$ being the corresponding exceptional divisor. We show that there is a weighted blowup correspondence between some certain absolute orbifold Gromov--Witten invariants of $\mathsf{X}$ relative to $\mathsf{S}$ and some certain relative orbifold Gromov--Witten invariants of the pair $({\mathsf{X}}_{\mathfrak{a}}|\mathsf{Z})$. As an application, we prove that the symplectic uniruledness of symplectic orbifold groupoids is a weighted blowup invariant.

arXiv comment50 pages; typos are fixed, the concept of symplectic uniruledness is modified; final version, to appear in Math. Ann; comments are welcome

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• Virtual neighborhood technique for pseudo-holomorphic spheres
• Relative orbifold Gromov-Witten theory and degeneration formula
• Rationally connected $3$-folds and symplectic geometry
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