Quantum Lefschetz hyperplane theorem

By Yuan-Pin Lee. 2000

57 citations recorded by the source corpus.

Abstract.The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov–Witten invariants of X and Gromov–Witten invariants of complete intersections Y in X is established.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ab096195ca6028d77b154560f33943aef09e315e

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Citations in the corpus

A desingularization of the main component of the moduli space of genus-one stable maps into ℙn (2006, methodology): “The complexity of this computation increases quickly with the degree d , but it has been completed in full generality in a number of different ways by Bertram [4], Gathmann [7], Givental [8], Lee [12] and Lian–Liu–Yau [14].”
On the genus-one Gromov-Witten invariants of complete intersections (2005, methodology): “This has been done for every d in [Ber], [Ga1], [Gi], [ Le ], and [LLY].”
Relative Gromov-Witten invariants and the mirror formula (2000, background): “This so-called \mirror formula" (also denoted \quantum Lefschetz hyperplane theorem" by some authors) has already been known for some time ([Gi], [LLY], [K], [B], [ L ]).”
Standard versus reduced genus-one Gromov–Witten invariants (2007, background): “lization theorem of [ABo]. The complexity of this computation increases quickly with the degree d, but it has been completed in full generality in a number of diﬀerent of ways; see [Ber], [Ga], [Gi], [Le], and [LLY]. If n=4, so Y is a threefold, then GWY 1,k(d;ψ) = d(a−5)+2 24 GWY 0,k(d;ψ) + ψ· e(π∗”
The reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces (2007, methodology): “At the end of this subsection we briey recall Givental’s approach [Gi] to dealing with this complexity and describe its interplay with our approach in genus 1; other proofs of (0.4) can be found in [Ber], [Ga], [ Le ], and [LLY].”
Symplectic geometry of Frobenius structures (2003, background): “) To make T a Frobenius manifold one takes the volume form ω to be 4The initial proof [21, 22] of the mirror formulas for toric complete intersections was based on fixed point localization while the new approaches [36, 16, 7] seem to be free of this restrictive technique.”
Floer theory for negative line bundles via Gromov–Witten invariants (2011, methodology): “This result, and Theorem 1, can be heuristically viewed as a symplectic analogue of the quantum Lefschetz hyperplane theorem [14]: the invariants of the hyperplane section B ⊂ M are recovered from invariants of the ambient M and a quantum multiplication operation by an Euler class.”
Quantum Riemann–Roch, Lefschetz and Serre (2001, background): “Corollary 6 is more general than the (otherwise similar) quantum Lefschetz hyperplane section theorems by Bertram and Lee [10,33] and Gathmann [18] for (i) it is applicable to arbitrary complete intersections Y without the restriction c1(Y ) ≥ 0, and (ii) it describes the J-functions not only over t”
Gromov-Witten invariants for abelian and nonabelian quotients (2004, background): “(4.3) Remark. As in the usual quantum Lefschetz hyperplane theorem ([Giv1], [Kim2], [ Lee ], [CG]), the change of variable is read off the asymptotics of the 1/~ expansion of the function I. In particular, if X//T is Fano, and for every effective curve C ⊂ X//T the intersection number C � (c1(X//T) ”