Relative quantum cohomology under birational transformations

Abstract

We study how relative quantum cohomology, defined in [TY20b] and [FWY20], varies under birational transformations. For toric complete intersections with simple normal crossings divisors that contain the loci of indeterminacy, we prove that their respective relative I-functions can be directly identified. For toric complete intersections with smooth divisors, we prove that their respective relative I-functions are related by analytic continuation. We also study some connections with extremal transitions and FJRW theory. Contents 1. Introduction 1.1. Motivations 1.2. Set-up 1.3. Results 1.4. Connection to other works 1.5. Future directions Acknowledgements 2. Relative quantum cohomology 2.1. Orbifold Gromov-Witten theory 2.2. Gromov-Witten theory of root stacks 2.3. Relative quantum cohomology 2.4. Givental formalism and mirror theorem 2.5. Relative quantum D-module 3. Set-up 4. Toric birational transformations 4.1. Toric set-up 4.2. Wall crossing 5. Relative quantum cohomology for toric pairs 5.1. The I-functions 5.2. The H-functions 5.3. Examples 5.4. The extended I-functions 6. Toric complete intersections 6.1. The standard set-up Date: April 4, 2022. 1 2 FENGLONG YOU 6.2. Exchanging the role of divisors 31 7. Relative quantum cohomology with non-toric divisors 36 7.1. Via local-orbifold correspondence 36 7.2. Via the hypersurface construction 40 8. Connection to transitions 44 9. Connection to FJRW theory 47 9.1. Example: cubic surface 48 9.2. Fano hypersurfaces in weighted projective spaces 50 References 53

A Gromov–Witten Theory for Simple Normal-Crossing Pairs Without Log Geometry

“…Then, one can compute punctured invariants through corresponding invariants of X D,∞ . Recently, the birational invariance of orbifold invariants has been investigated in [8,41].…”