Orbifold Gromov-Witten theory of weighted blowups

Chen, Bohui; Du, Cheng-Yong; Wang, Rui

doi:10.1007/s11425-020-1774-x

Cited by 5 publications

(5 citation statements)

References 35 publications

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“…The moduli space F e = F ([g e ], κ e ) is a fibration over D ′ [ge] with fiber being the (1 + 0 + 1)-pointed, genus zero relative moduli space of P r,1 (C ⊕ C) ⋊ g e , whose topological type is determined by the topological type of F e (cf. [9]). The D ′…”

Section: And (37)mentioning

confidence: 99%

“…A cycle version of Leray-Hirsch result. As an application of the computation of DR-cycles for L → D we could prove a cycle version of Leray-Hirsch result for orbifold Gromov-Witten theory obtained in [9] under the assumption that D is a quotient orbifold of a smooth quasi-projective scheme by a linear algebraic group. Theorem 3.17.…”

Section: 3mentioning

confidence: 99%

“…Hence for every matched pair of summands in (4.6) and (4.7), we have Γ(r) − = Γ − and Γ(r) + is obtained from Γ + via the convention in §4.1.3, see (4. to denote topological data of relative stable maps to (D 0 |Y|D ∞ ), where We have proved in [9] the following result. We will find out that Lemma 4.4 is not enough to prove Lemma 4.3; we need to consider the more general case that [h 1 ] satisfies ρ([h 1 ]) = 1.…”

Section: Lemma 42mentioning

confidence: 99%

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Double ramification cycles with orbifold targets

Chen¹,

Du²,

Wang³

2020

Preprint

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Section: And (37)mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Lemma 42mentioning

confidence: 99%

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Double ramification cycles with orbifold targets

Chen¹,

Du²,

Wang³

2020

Preprint

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“…Early work here focused on blow-ups in points and on exploiting structural properties of quantum cohomology such as the Witten--Dijkgraaf--Verlinde--Verlinde (WDVV) equations and reconstruction theorems [Gat96, GP98, Gat01]. Subsequent approaches used symplectic methods pioneered by Li–Ruan [LR01, HLR08, Hu00, Hu01], or the Degeneration Formula following Maulik–Pandharipande [MP06, HHKQ18, CDW20], or a direct analysis of the moduli spaces involved and virtual birationality arguments [Man12, Lai09, AW18]. In each case, the aim was to prove ‘birational invariance’: that certain specific Gromov–Witten invariants remain invariant under blow-up.…”

Section: Introductionmentioning

confidence: 99%

The Abelian/Nonabelian Correspondence and Gromov–Witten Invariants of Blow-Ups

Coates

Wendelin

Qaasim

2022

Forum of Mathematics, Sigma

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We prove the abelian/nonabelian correspondence with bundles for target spaces that are partial flag bundles, combining and generalising results by Ciocan-Fontanine–Kim–Sabbah, Brown, and Oh. From this, we deduce how genus-zero Gromov–Witten invariants change when a smooth projective variety X is blown up in a complete intersection defined by convex line bundles. In the case where the blow-up is Fano, our result gives closed-form expressions for certain genus-zero invariants of the blow-up in terms of invariants of X. We also give a reformulation of the abelian/nonabelian Correspondence in terms of Givental’s formalism, which may be of independent interest.

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“…For example, The natural map from the root stack X D,r to X is birational. It was proved in [TY16] and [CDW20] that the Gromov-Witten theory of X D,r is determined by the Gromov-Witten theory of X, Gromov-Witten theory of D and the restriction map H * (X) → H * (D). On the other hand, the relation between relative Gromov-Witten theory of (X D,r , D r ) and (X, D) is significantly simpler: they coincide by [AF16].…”

mentioning

confidence: 99%

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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Orbifold Gromov-Witten theory of weighted blowups

Cited by 5 publications

References 35 publications

Double ramification cycles with orbifold targets

Double ramification cycles with orbifold targets

The Abelian/Nonabelian Correspondence and Gromov–Witten Invariants of Blow-Ups

Relative quantum cohomology under birational transformations

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