Gromov–Witten invariants of a quintic threefold and a rigidity conjecture

Li, Jun; Zinger, Aleksey

doi:10.2140/pjm.2007.233.417

Cited by 8 publications

(7 citation statements)

References 17 publications

(29 reference statements)

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“…Theorem 1.3 is a particular case of [30] which holds for X any compact symplectic manifold of dimension 2 and 3 and of [29] which holds for X any compact symplectic manifold of any dimension. A similar statement appears in [19,20]. Algebraically it has been proved by Chang and Li [5], for X the quintic threefold, using a slightly different definition for reduced Gromov-Witten invariants to Zinger [30].…”

Section: Introduction Outline Of the Proofmentioning

confidence: 57%

“…Reduced invariants were defined, using symplectic methods, and compared to Gromov-Witten invariants by Zinger [28][29][30]32]. Li-Zinger showed [19,20] that reduced Gromov-Witten invariants are the integral of the top Chern class of a sheaf over the main component of M (P); this is an analog, for reduced genus 1 invariants, of the quantum Lefschetz hyperplane property [19,20]. In view of [30] this also gives a proof of Theorem 1.3.…”

Section: Introduction Outline Of the Proofmentioning

confidence: 99%

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A splitting of the virtual class for genus one stable maps

Coates,

Manolache

2018

Preprint

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spaces of stable maps to a smooth projective variety typically have several components. We express the virtual class of the moduli space of genus one stable maps to a smooth projective variety as a sum of virtual classes of the components. The key ingredient is a generalised functoriality result for virtual classes. We show that the natural maps from 'ghost' components of the genus one moduli space to moduli spaces of genus zero stable maps satisfy the strong push forward property. As a consequence, we give a cycle-level formula which relates standard and reduced genus one Gromov-Witten invariants of a smooth projective Calabi-Yau theefold. CÃ ]. This proves (i).

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Section: Introduction Outline Of the Proofmentioning

confidence: 57%

Section: Introduction Outline Of the Proofmentioning

confidence: 99%

A splitting of the virtual class for genus one stable maps

Coates,

Manolache

2018

Preprint

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“…Various special cases of Theorem 1.2, such as those in Examples A-C, are standard in the algebraic setting and are used in [3], [13], and [26], for example. Some special cases of Theorem 1.2 have appeared in the symplectic setting as well, including in [16], [24], and [33]. Examples B and C generalize Example A in two opposite directions.…”

Section: Introductionmentioning

confidence: 99%

A comparison theorem for Gromov–Witten invariants in the symplectic category

Zinger

2011

Advances in Mathematics

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We exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare Gromov-Witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. Various special cases of the comparison theorem in this paper have long been used in the algebraic category; some of them have also appeared in the symplectic setting. Combined with the inherent flexibility of the symplectic category, the main theorem leads to a confirmation of Pandharipande's Gopakumar-Vafa prediction for GW-invariants of Fano classes in 6-dimensional symplectic manifolds. The proof of the main theorem uses deformations of the Cauchy-Riemann equation that respect the submanifold and Carleman Similarity Principle for solutions of perturbed Cauchy-Riemann equations. In a forthcoming paper, we apply a similar approach to relative Gromov-Witten invariants and the absolute/relative correspondence in genus 0. * Partially supported by a Sloan fellowship and DMS Grant 0604874 A comparison theorem for GW-invariantsWe will denote byZ + the set of non-negative integers. Let (X, ω) be a compact symplectic manifold. If g ∈Z + , S is a finite set, β ∈ H 2 (X; Z), and J is an ω-tame 1 almost complex structure on X, denote by M g,S (X, β; J) the moduli space of equivalence classes of stable S-marked genus g degree β J-holomorphic maps to X. For each j ∈ S, there is a well-defined evaluation map ev j : M g,S (X, β; J) −→ X.(1.1)As standard in GW-theory, we will denote by ψ j ∈ H 2 M g,S (X, β; J)the first chern class of the universal cotangent line bundle for the j-th marked point. The space M g,S (X, β; J) carries a natural VFC, which is independent of J and will be denoted by 1 an almost complex structure on (X, ω) is ω-tame if ω(v, Jv) > 0 for all v ∈ T X with v = 0 hood of Mg,S(X, β; J) in the space of equivalence classes of L p 1 -maps to X; there are well-defined evaluation maps evj and cohomology classes ψj on this space as well. 4 We can assume that this is possible, since each κj can be replaced by a multiple for our purposes.

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“…where f : M β,1 −→ M β is the forgetful map and N |β| is the normal bundle to the family of simple curves of class |β|. Similarly to Section 3.3 in [12], we choose a family of "exponential" maps…”

Section: Preliminariesmentioning

confidence: 99%

“…12) where O B −→ Z T ,B is the obstruction bundle associated with the moduli space M (0,m) (X, β).Let ν ∈ Γ(Z T , O) be the section induced by ν:ν(b) is the projection of ν(b) to the cokernel of D b . We writeν There is a natural projection mapπ : Z T ,B −→ M β,1 , sending π B ([Σ, u]) to (u(Σ), u(Σ P )).…”

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confidence: 99%

Enumerative geometry of Calabi–Yau 5-folds

Pandharipande¹,

Zinger²

Advanced Studies in Pure Mathematics

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Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 5-folds. We find recursions for meeting numbers of genus 0 curves, and we determine the contributions of moving multiple covers of genus 0 curves to the genus 1 Gromov-Witten invariants. The resulting invariants, conjectured to be integral, are analogous to the previously defined BPS counts for Calabi-Yau 3 and 4-folds. We comment on the situation in higher dimensions where new issues arise.Two main examples are considered: the local Calabi-Yau P 2 with normal bundle ⊕ 3 i=1 O(−1) and the compact Calabi-Yau hypersurface X 7 ⊂ P 6 . In the former case, a closed form for our integer invariants has been conjectured by G. Martin. In the latter case, we recover in low degrees the classical enumeration of elliptic curves by Ellingsrud and Strömme.1 A nonsingular curve E ⊂ X with normal bundle NE is super-rigid if, for every dominant stable map f : C → E, the vanishing H 0 (C, f * NE) = 0 holds. *(1.1) be the component projection maps.Condition 1 If u : P 1 −→ X is a simple holomorphic map, H 1 P 1 , u * T X) = 0.By Condition 1, M * 0,J (X, β) is a nonsingular variety of the expected dimension 2+|J|. * 0,1 (X, β) −→ X is an immersion. If β 1 , β 2 ∈ H + (X) and J 1 , J 2 are finite sets, M * 0,(J 1 ,J 2 ) X, (β 1 , β 2 ) is smooth of dimension 1+|J 1 |+|J 2 | and consists of simple maps. * 0,(J 1 ,J 2 ) X, (β 1 , β 2 ) is the same.Conditions 1-4 can be extended to define an ideal Calabi-Yau n-fold for any n. However, Lemma 1.1, which depends on the dimension counting argument in the preceding paragraph, does

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Gromov–Witten invariants of a quintic threefold and a rigidity conjecture

Cited by 8 publications

References 17 publications

A splitting of the virtual class for genus one stable maps

A splitting of the virtual class for genus one stable maps

A comparison theorem for Gromov–Witten invariants in the symplectic category

Enumerative geometry of Calabi–Yau 5-folds

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