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

Abstract

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

On weighted-blowup formulae of genus zero orbifold Gromov–Witten invariants

“…In fact, is the weight- projectivization (cf. [CDH19]) of , and the normal line bundle of in is , the pull back of the tautological line bundle via the natural projection .…”

“…Let be the weight- blowup (cf. [MM12, CDH19]) of along with weight Denote by the exceptional divisor of , which is the weight- projectivization of .…”

“…See, for example, [ALR07] and [MP97]. One can see also [CDH19, § 2] for a brief introduction to orbifold groupoids, Chen–Ruan cohomology, weighted blowups, etc. The strategy to prove the weighted-blowup formulae is exactly the same as the proof of the blowup formula in § 2.…”

Double ramification cycles with orbifold targets

“…See for example Adem-Leida-Ruan ( [5]) and ). One can see also [8,Section 2] for a brief introduction of orbifold groupoids and Chen-Ruan cohomology etc.…”

“…See[8, Lemma 5.6] for the form of orbifold maps between cyclic group quotients of weighted projective spaces.…”