Orbifold Gromov--Witten theory of weighted blowups

Abstract

Consider a compact symplectic sub-orbifold groupoid $\mathsf{S}$ of a compact symplectic orbifold groupoid $(\mathsf{X},\omega )$. Let ${\mathsf{X}}_{\mathfrak{a}}$ be the weight-$\mathfrak{a}$ blowup of $\mathsf{X}$ along $\mathsf{S}$, and ${\mathsf{D}}_{\mathfrak{a}}={\mathsf{PN}}_{\mathfrak{a}}$ be the exceptional divisor, where $\mathsf{N}$ is the normal bundle of $\mathsf{S}$ in $\mathsf{X}$. In this paper we show that the absolute orbifold Gromov--Witten theory of ${\mathsf{X}}_{\mathfrak{a}}$ can be effectively and uniquely reconstructed from the absolute orbifold Gromov--Witten theories of $\mathsf{X}$, $\mathsf{S}$ and ${\mathsf{D}}_{\mathfrak{a}}$, the natural restriction homomorphism $H_{\text{CR}}^{\ast }(\mathsf{X})\to H_{\text{CR}}^{\ast }(\mathsf{S})$ and the first Chern class of the tautological line bundle over ${\mathsf{D}}_{\mathfrak{a}}$. To achieve this we first prove similar results for the relative orbifold Gromov--Witten theories of $({\mathsf{X}}_{\mathfrak{a}}|{\mathsf{D}}_{\mathfrak{a}})$ and $({\mathsf{N}}_{\mathfrak{a}}|{\mathsf{D}}_{\mathfrak{a}})$. As applications of these results, we prove an orbifold version of a conjecture of Maulik--Pandharipande on the Gromov--Witten theory of blowups along complete intersections, a conjecture on the Gromov--Witten theory of root constructions and a conjecture on Leray--Hirsch result for orbifold Gromov--Witten theory of Tseng--You.

44pages; some typos fixed; comments are welcome; accepted for publication in SCIENCE CHINA Mathematics

References in corpus (7)

• Birational cobordism invariance of uniruled symplectic manifolds
• Gromov-Witten theory of banded gerbes over schemes
• Symplectic virtual localization of Gromov-Witten invariants
• Rationally connected $3$-folds and symplectic geometry
• Chern classes and Gromov--Witten theory of projective bundles
• Double ramification cycles with orbifold targets
• Gluing principle for orbifold stratified spaces