Quantum Lefschetz hyperplane theorem
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 generating genus zero functions of Gromov-Witten invariants, \(J^E_X\) and \(I^E_X\), actually coincide. From these two main results, whose proof is based on equivariant Gromov-Witten theory and virtual localization techniques, the authors deduce a fundamental relationship between the Gromov-Witten invariants of the variety \(X\) and those of a complete intersection \(Y\) in \(X\). This relationship shows that enumerative information on \(Y\) can be derived from the one on \(X\), and that in a way generalizing the quantum Lefschetz hyperplane theorem à la Givental-Kim. The significance of the author's generalized approach is demonstrated by the fact that some of the already established versions of the mirror theorem can be rediscovered in this context. As to the reconstruction theorems for Gromov-Witten invariants used in the course of the paper, the authors rely on separate recent results by \textit{Y.-P. Lee} and \textit{R. Pandharipande} [Am. J. Math. 126, No. 6, 1367--1379 (2004; Zbl 1080.14065)].
scientific article; zbMATH DE number 1655743