Invariance of Gromov–Witten theory under a simple flop

Iwao, Yoshihiro; Lee, Y.-P.; Lin, Hui-Wen; Wang, Chin-Lung

doi:10.1515/crelle.2011.097

Cited by 14 publications

(19 citation statements)

References 14 publications

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“…Note that it has been independently discovered and proved by Iwao, Lee, Lin, and Wang [6]. We remark that our proof is different from theirs and seems simpler.…”

Section: Gromov-witten Invariantsmentioning

confidence: 63%

“…We remark that our proof is different from theirs and seems simpler. However, [6] contains a more general result.…”

Section: Gromov-witten Invariantsmentioning

confidence: 99%

“…On the other hand, in [6,8], the degree zero theories and the genus zero theories of X and X have been determined, and the relations k = −1 r−k k k≥ 1 have also been established. Therefore, our statement follows from Theorem 0.1 and a direct comparison of both sides of (1.2).…”

Section: Exceptional N-point Functionsmentioning

confidence: 99%

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A Note on a Generalization of the Multiple Cover Formula and the Simple Flop

Cheong

2012

Communications in Algebra

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“…Note that it has been independently discovered and proved by Iwao, Lee, Lin, and Wang [6]. We remark that our proof is different from theirs and seems simpler.…”

Section: Gromov-witten Invariantsmentioning

confidence: 63%

“…We remark that our proof is different from theirs and seems simpler. However, [6] contains a more general result.…”

Section: Gromov-witten Invariantsmentioning

confidence: 99%

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A Note on a Generalization of the Multiple Cover Formula and the Simple Flop

Cheong

2012

Communications in Algebra

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“…al. [29,27,28]. Finally, the most relevant form of the conjecture for our purposes was proposed by Coates-Iritani-Tseng [15] using physical considerations from mirror symmetry.…”

Section: Complete Intersectionsmentioning

confidence: 99%

Gromov–Witten Theory of Toric Birational Transformations

Acosta

Shoemaker

2019

International Mathematics Research Notices

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We investigate the effect of a general toric wall crossing on genus zero Gromov-Witten theory. Given two complete toric orbifolds X + and X − related by wall crossing under variation of GIT, we prove that their respective I-functions are related by linear transformation and asymptotic expansion. We use this comparison to deduce a similar result for birational complete intersections in X + and X − . This extends the work of the previous authors in [2] to the case of complete intersections in toric varieties, and generalizes some of the results of Coates-Iritani-Jiang [14] on the crepant transformation conjecture to the setting of non-zero discrepancy.

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“…This is the most studied case in the existing literature: Theorem 2.1 (Li-Ruan, 1997 [5] In fact the invariance for the total ancestror potential in all genera is established in [2] based on results in [3]. Recently the authors studied the problem for general ordinary flops without the "simple" assumption [4] and obtained, among other things, the invariance of quantum rings under the split assumptions of the bundles F and F .…”

Section: Ordinary Flopsmentioning

confidence: 99%

Analytic continuations of quantum cohomology

Ji¹,

Poon²,

Yang³

et al. 2012

Fifth International Congress of Chinese Mathematicians

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Abstract. In the study of local model of ordinary flops, we introduce the Birkhoff factorizaton procedure to produce the generalized mirror map, which is the essential ingredient in performing analytic continuations of quantum cohomology in the Kähler moduli spaces. We survey this procedure in this article and provide an example to illustrate its validity. GW theoryThis article consists of the second author's talk at ICCM 2010 at Beijing under the same title, with more details filled in. The purpose is to introduce the results obtained by the authors in [3] and [4] and to work out a typical example.Let X be a projective manifold over C and M g,n (X, β) be the moduli space of n-pointed stable maps f : (C; p 1 , . . . , p n ) → X from a nodal curve C with arithmetic genus p a (C) = g, [f (C)] = β ∈ NE(X), the Mori cone of effective one cycles.Li-Tian and Behrend-Fantechi had constructed the virtual fundamental classAnd then the (descendent) Gromov-Witten invariants are defined by integration along these cycles: For a i ∈ H(X), The (primary) GW potential is the generating function of all GW invariants without descendent insertions (k i = 0). Let {T i } be a cohomology basis of H = H(X), {T i } be its dual basis, and t = t i T i ∈ H be a general element with t i being the coordinates. Let {q β }'s be the formal (Novikov) variables. Then

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Invariance of Gromov–Witten theory under a simple flop

Abstract: We show that the generating functions of Gromov-Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. This is a sequel to [14].

Cited by 14 publications

References 14 publications

A Note on a Generalization of the Multiple Cover Formula and the Simple Flop

A Note on a Generalization of the Multiple Cover Formula and the Simple Flop

Gromov–Witten Theory of Toric Birational Transformations

Analytic continuations of quantum cohomology

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