Quantum hyperplane section theorem for homogeneous spaces
type: Item
instance of: scholarly article
title: Quantum hyperplane section theorem for homogeneous spaces
zbMATH Open document ID: 1023.14028
DOI: 10.1007/BF02392947
author: Bumsig Kim
published in: Acta Mathematica
publication date: 2000-12-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/alg-geom/9712008
review text: The author formulates and proves a certain mirror theorem for complete intersections in homogeneous spaces. Starting from the previously known fact that for some large class of Calabi-Yau manifolds, namely for complete intersections in toric manifolds or homogeneous spaces, the mirror symmetry conjecture can be interpreted as a quantum cohomology counterpart of the so-called weak Lefschetz hyperplane section theorem, the author investigates more closely this associated ``quantum hyperplane section conjecture'' for homogeneous spaces. When the ambient space is a symplectic toric manifold, then this weak hyperplane section conjecture is indeed true, according to A. Givental's mirror theorem for such manifolds [cf. \textit{A. Givental} in: Topological field theory, primitive forms and related topics, Proc. 38th Taniguchi Symp. and RIMS Symp., Kyoto 1996; Prog. Math. 160, 141-175 (1998; Zbl 0936.14031)]. In the present work, the author extents Givental's approach, in a very subtle way, to the case where the ambient space \(X\) is a compact homogeneous space of a semisimple complex Lie group \(G\), ending up with a proof of the weak hyperplane section conjecture as formulated in the foregoing paper ``Mirror symmetry and toric degenerations of partial flag manifolds'' by \textit{V. Batyrev}, \textit{I. Ciocan-Fontaine}, \textit{B. Kim} the author and \textit{D. van Straten} Roughly speaking, the author's mirror theorem for homogeneous spaces states that there is some explicit relation between quantum differential equations of the ambient manifold \(X\) and those of a non-negative, smooth zero locus of a certain convex decomposable vector bundle \(V\) over \(X\). As a consequence, it follows that the quantum cohomology ring of such a homogeneous space \(X\) enjoys a specific functoriality property with regard to embeddings of semi-positive complete intersections, which is highly remarkable. Also, by applying the main result of his work, the author is able to confirm the pr…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14J32
zbMATH DE Number: 1541211
zbMATH Keywords: Grassmannians
zbMATH Keywords: quantum hyperplane section conjecture
zbMATH Keywords: complete intersections
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: mirror theorem
zbMATH Keywords: quantum cohomology ring
MaRDI profile type: Publication
arXiv ID: alg-geom/9712008
cites work: Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians
cites work: Mirror symmetry and toric degenerations of partial flag manifolds
cites work: Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties
cites work: Gromov-Witten invariants in algebraic geometry
cites work: The intrinsic normal cone
cites work: Stacks of stable maps and Gromov-Witten invariants
cites work: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory.
cites work: Notes on stable maps and quantum cohomology
cites work: Homological geometry. I: Projective hypersurfaces
cites work: Equivariant Gromov - Witten Invariants
cites work: Elliptic Gromov-Witten invariants and the generalized mirror conjecture
cites work: Quantum cohomology of flag manifolds \(G/B\) and quantum Toda lattices.
cites work: Q4858415
cites work: Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties
cites work: Mirror principle. I
cites work: IRREDUCIBILITY OF $\overline{M}_{0,n}(G/P,\beta)$
OpenAlex ID: W2059343253
description: scientific article