Quantum Lefschetz hyperplane theorem
type: Item
instance of: scholarly article
title: Quantum Lefschetz hyperplane theorem
zbMATH Open document ID: 1082.14056
author: Y.-P. Lee
published in: Inventiones Mathematicae
publication date: 2001-01-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0003128
review text: Let \(X\subset\mathbb{P}^N\) be a complex projective variety of dimension \(n\), and let \(Y=\mathbb{P}^{N-1}\cap X\) a hyperplane section such that \(U= X\setminus Y\) is smooth and \(n\)-dimensional. Then the classical Lefschetz hyperplane theorem asserts that the restriction morphism \(H^k(X,\mathbb{Z})\to H^k(Y,\mathbb{Z})\) is an isomorphism for \(k< n\) and is still injective for \(k= n-1\). In the framework of quantum cohomology, an analoguous principle has been suggested by \textit{A. B. Givental} [Int. Math. Res. Not. 1996, No. 13, 613--663 (1996; Zbl 0881.55006)] and mathematically formalized by \textit{B. Kim} [Acta Math. 183, No. 1, 71--99 (1999; Zbl 1023.14028)], where the so-called quantum Lefschetz hyperplane section principle was formulated for complete intersections in a generalized flag variety. In this setting, the quantum Lefschetz hyperplane theorem is basically equivalent to the mirror theorem for this class of varieties, and a generalization of the quantum Lefschetz hyperplane theorem to a wider class of varieties would therefore imply a generalization of the mirror theorem to that class, too. In the paper under review, the author provides a generalization of the quantum Lefschetz hyperplane theorem to arbitrary complete intersections \(Y\) in a smooth complex projective variety \(X\) and therefore a generalized mirror theorem as well. More precisely, let \(X\) be a smooth projective variety embedded in \(\prod^N_{i=1} \mathbb{P}^{r_i}= P\), and let \(E\) be a vector bundle on \(X\) which is decomposed into a direct sum of pull-backs of convex and concave line bundles on \(P\) such that some nonnegativity condition for its first Chern class holds. Then there are two formal functions \(J^E_X\) and \(I^E_X\) with values in the cohomology ring of \(X\) and the author's first main theorem states that these two functions are equivalent up to mirror transformation. The second main theorem asserts under which conditions on the bundle \(E\) these two …
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14D21
zbMATH DE Number: 1655743
zbMATH Keywords: Gromov-Witten invariants
zbMATH Keywords: quantum cohomology
zbMATH Keywords: mirror symmetry
zbMATH Keywords: mirror conjecture
zbMATH Keywords: complete intersections
zbMATH Keywords: Lefschetz principle
MaRDI profile type: Publication
OpenAlex ID: W3100979339
arXiv ID: math/0003128
DOI: 10.1007/S002220100145
description: scientific article; zbMATH DE number 1655743