2023/05/18 by Georg Oberdieck, Oberdieck, Georg
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2305.11115
openalex publication_date 2023/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We determine the Gromov-Witten invariants of the local Enriques surfaces for all genera and curve classes and prove the Klemm-Mariño formula. In particular, we show that the generating series of genus 1 invariants of the Enriques surface is the Fourier expansion of a certain power of Borcherds automorphic form on the moduli space of Enriques surfaces. We also determine all Vafa-Witten invariants of the Enriques surface. The proof uses the correspondence between Gromov-Witten and Pandharipande-Thomas theory. On the Gromov-Witten side we prove the relative Gromov-Witten potentials of an elliptic Enriques surfaces are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. On the sheaf side, we relate the Pandharipande-Thomas invariants of the Enriques-Calabi-Yau threefold in fiber classes to the 2-dimensional Donaldson-Thomas invariants by a version of Toda's formula for local K3 surfaces. Altogether, we obtain sufficient modular constraints to determine all invariants from basic geometric computations.