Singular symplectic flops and Ruan cohomology

TL;DR

This paper investigates the symplectic geometry of singular orbifold conifolds, constructs orbifold symplectic flops, and proves the invariance of Ruan cohomology under these flops, confirming Ruan's conjecture.

Contribution

It introduces the study of orbifold symplectic flops on singular conifolds and proves the invariance of Ruan cohomology in this context, extending previous results.

Findings

Orbifold symplectic flops preserve Ruan cohomology.

Construction of orbifold symplectic conifold transitions and flops.

Verification of Ruan's conjecture for a class of singular orbifolds.

Abstract

In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient $$ W_r=\{(x,y,z,t)|xy-z^{2r}+t^2=0 \}/\mu_r(a,-a,1,0), r\geq 1, $$ which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let $X$ and $Y$ be two symplectic orbifolds connected by such a flop. We study orbifold Gromov-Witten invariants of exceptional classes on $X$ and $Y$ and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.