Extremal transitions via quantum Serre duality

Mi, Rongxiao; Shoemaker, Mark

doi:10.48550/arxiv.2006.09907

Cited by 1 publication

(7 citation statements)

References 31 publications

(74 reference statements)

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“…Therefore, we just need to compare these local invariants. Then it becomes similar to the approach in [MS20] where the authors study Gromov-Witten theory under extremal transitions using quantum Serre duality. The varieties related by transitions in [MS20] are hypersurfaces in toric varieties where the ambient toric varieties are related by toric blow-ups.…”

Section: Case (1)mentioning

confidence: 88%

“…We focus on the type of toric birational transformations called discrepant resolutions. With a mild generalization of [MS20] to discrepant resolutions, we prove the following statement in terms of quantum D-module.…”

Section: Case (1)mentioning

confidence: 92%

“…Transition. In [MS20], the authors compared genus zero Gromov-Witten theory of toric hypersurfaces related by extremal transitions arising from toric blow-ups. The quantum D-modules are related by analytic continuations and specializations of parameters.…”

Section: Connection To Other Workmentioning

confidence: 99%

“…Then D and D are related by transitions. The main result of [MS20] (see also [Mi17]) can be stated in terms of I-functions as follows. There is an explicit degree-preserving linear transformation L ∶ H * ( D) → H * (D) such that…”

Section: Connection To Other Workmentioning

confidence: 99%

“…We would like to choose a i,± such that Note that, what we consider here is slightly more general than what was considered in [MS20], where they consider the case of blow-ups. In here, we include the cases when φ ∶ X + / / ❴ ❴ ❴ X − is a discrepant resolution in general (including a weighted blow-up).…”

Section: Relative Quantum Cohomology With Non-toric Divisorsmentioning

confidence: 99%

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Relative quantum cohomology under birational transformations

You¹

2022

Preprint

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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

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Section: Case (1)mentioning

confidence: 88%

Section: Case (1)mentioning

confidence: 92%

Section: Connection To Other Workmentioning

confidence: 99%

Section: Connection To Other Workmentioning

confidence: 99%

Section: Relative Quantum Cohomology With Non-toric Divisorsmentioning

confidence: 99%

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