Quantum Lefschetz hyperplane theorem

Lee, Y.-P.

doi:10.1007/s002220100145

Cited by 40 publications

(47 citation statements)

References 19 publications

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“…Quantization of the pq-terms yields the linear vector field associated to the linear map q(z) → −[ch k+1 (E)q(z)/z] + , whilst the q 2 -term −(ch k+1 (E)q 0 , q 0 )/2 matches the second summand in (35) due to (vii) and (6). Evaluating the third summand using (ix) we conclude that the terms in (33) involving B 0 can be written as…”

Section: Appendix 1 the Proof Of Theoremmentioning

confidence: 88%

“…What has been missing so far is the part that Birkhoff factorization plays in the formulations. Now restricting J X,Y and I X,Y to the small parameter space H ≤2 (X, Λ) and assuming that c 1 (E) ≤ c 1 (X) we can derive the quantum Lefschetz theorems of [4], [9], [18], [29], [33]. A dimensional argument shows that the series I X,Y on the small parameter space has the form…”

Section: Mirror Formulasmentioning

confidence: 99%

“…It is described in terms of the stationary phase asymptotics a ρ (z) of the oscillating integral Assuming E to be a line bundle, we derive a "quantum hyperplane section theorem" (Theorem 2). It is more general than the earlier versions [4], [29], [18], [33] in the sense that the restrictions t ∈ H ≤2 (X; Q) on the space of parameters and c 1 (E) ≤ c 1 (X) on the Fano index are removed.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Riemann–Roch, Lefschetz and Serre

2007

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Given a holomorphic vector bundle E over a compact Kähler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f : Σ → X with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle H 0 (Σ, f * E) H 1 (Σ, f * E). Using the formalism of quantized quadratic Hamiltonians [25], we express the descendant potential for the twisted theory in terms of that for X. This result (Theorem 1) is a consequence of Mumford's Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants.When E is concave and the C × -equivariant inverse Euler class is chosen as the characteristic class, the twisted invariants of X give Gromov-Witten invariants of the total space of E. "Nonlinear Serre duality" [21], [23] expresses Gromov-Witten invariants of E in terms of those of the super-manifold ΠE: it relates Gromov-Witten invariants of X twisted by the inverse Euler class and E to Gromov-Witten invariants of X twisted by the Euler class and E * . We derive from Theorem 1 nonlinear Serre duality in a very general form (Corollary 2).When the bundle E is convex and a submanifold Y ⊂ X is defined by a global section of E, the genus-zero Gromov-Witten invariants of ΠE coincide with those of Y . We establish a "quantum Lefschetz hyperplane section principle" (Theorem 2) expressing genus-zero Gromov-Witten invariants of a complete intersection Y in terms of those of X. This extends earlier results [4], [9], [18], [29], [33] and yields most of the known mirror formulas for toric complete intersections.

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Section: Appendix 1 the Proof Of Theoremmentioning

confidence: 88%

Section: Mirror Formulasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum Riemann–Roch, Lefschetz and Serre

2007

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“…This is an abstract formulation of a very general result which in the limit λ → 0 allows one to compute genus 0 GW-invariants of a hypersurface in X defined by a section of E in a form generalizing Quantum Lefschetz Theorems of B. Kim, A. Bertram, Y.-P. Lee and A. Gathmann [2,3,35,16]. Namely, Theorem 4 applies beyond "small" quantum cohomology theory, to general type complete intersections as well, and also offers a new insight on the nature of mirror maps as a very special case of Birkhoff factorization in loop groups.…”

Section: Proposition the Ancestor Potentialsf τ (Defined In The Axiomentioning

confidence: 99%

Symplectic geometry of Frobenius structures

Givental¹

2004

Frobenius Manifolds

115

108

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The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in two-dimensional topological field theories "coupled to gravity at the tree level". The purpose of these notes is to reiterate and expand the viewpoint, outlined in the paper [7] of T. Coates and the author, which recasts this concept in terms of linear symplectic geometry and exposes the role of the twisted loop group L (2) GL N of hidden symmetries.We try to keep the text introductory and non-technical. In particular, we supply details of some simple results from the axiomatic theory, including a several-line proof of the genus 0 Virasoro constraints not mentioned elsewhere, but merely quote and refer to the literature for a number of less trivial applications, such as the quantum Hirzebruch-Riemann-Roch theorem in the theory of cobordism-valued Gromov-Witten invariants. The latter is our joint work in progress with Tom Coates, and we would like to thank him for numerous discussions of the subject.The author is also thankful to Claus Hertling, Yuri Manin and Matilde Marcolli for the invitation to the workshop on Frobenius structures held at MPIM Bonn in Summer 2002.Gravitational descendents. In Witten's formulation of topological gravity [43] one is concerned with certain "correlators" called gravitational descendents. The totality of genus 0 gravitational descendents is organized as the set of Taylor coefficients of a single formal function F of a sequence of vector variables t = (t 0 , t 1 , t 2 , ...). The vectors t i are elements of a finite-dimensional vector space 1 which we will denote H. One assumes that H is equipped with a symmetric nondegenerate bilinear form (·, ·) and with a distinguished non-zero element 1. The axioms are formulated as certain partial differential equations on F : an infinite set of Topological Recursion Relations (TRR), the String Equation (SE) and the Dilaton Equation (DE). To state the axioms using the following coordinate notation: introduce a basis {φ α } in H such thatand write ∂ α,k for the partial derivative ∂/∂t α k . Then the axioms read:Research is partially supported by NSF Grant DMS-0072658. 1 or super-space, but we will systematically ignore the signs that may come out of this possibility.

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“…It can be computed via the classical localization theorem of [Atiyah and Bott 1984]. The complexity of this computation increases quickly with the degree d, but a closed formula has been obtained in [Bertram 2000], [Gathmann 2002], [Givental 1999], [Lee 2001], and [Lian et al 1997].…”

Section: Introductionmentioning

confidence: 99%

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

Li¹,

Zinger²

2007

Pacific J. Math.

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We show that a widely believed conjecture concerning rigidity of genuszero and genus-one holomorphic curves in Calabi-Yau threefolds implies a relation between the genus-one GW-invariants of a quintic threefold in ‫ސ‬ 4 and the genus-zero and genus-one GW-invariants of ‫ސ‬ 4 . This relation is a special case of a general formula for the genus-one GW-invariants of complete intersections obtained in a previous paper. In contrast to the general case, this paper's derivation is more geometric and makes direct use of the rigidity property. Thus, it provides further evidence for the rigidity conjecture in low genera. On the other hand, this paper also suggests a potential way of disapproving the less commonly believed generalization of the rigidity conjecture to arbitrary genus.

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Quantum Lefschetz hyperplane theorem

Cited by 40 publications

References 19 publications

Quantum Riemann–Roch, Lefschetz and Serre

Quantum Riemann–Roch, Lefschetz and Serre

Symplectic geometry of Frobenius structures

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

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