2023/08/01 by Deniz Genlik, Hsian-Hua Tseng, Hsian‐Hua Tseng +2
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory
paper · pdf · doi:10.1016/j.aim.2026.111050
We study the structure of the higher genus Gromov-Witten theory of the total space Kℙn-1 of the canonical bundle of the projective space ℙn-1. We prove the finite generation property for the Gromov-Witten potential of Kℙn-1 by working out the details of its cohomological field theory (CohFT). More precisely, we prove that the Gromov-Witten potential of Kℙn-1 lies in an explicit polynomial ring using the Givental-Teleman classification of the semisimple CohFTs. In arXiv:2301.08389, we carried out a parallel study for [ℂn/ℤn] and proved that the Gromov-Witten potential of [ℂn/ℤn] lies in a similar polynomial ring. The main result of this paper is a crepant resolution correspondence for higher genus Gromov-Witten theories of Kℙn-1 and [ℂn/ℤn], which is proved by establishing an isomorphism between the polynomial rings associated to Kℙn-1 and [ℂn/ℤn]. This paper generalizes the works of Lho-Pandharipande arXiv:1804.03168 for the case of [ℂ3/ℤ3] and Lho arXiv:2211.15878 for the case [ℂ5/ℤ5] to arbitrary n≥ 3.