Quantum hyperplane section theorem for homogeneous spaces
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 predictions for Calabi-Yau complete intersections in Grassmannians, which he had made with V. Batyrev et al. in the above-mentioned, foregoing paper published in 1998. All together, the article consists of eight very detailed sections, including a just as thorough introduction, and is in regard of its comprehensiveness and clarity nearly self-contained. The results certainly represent a major contribution towards the understanding of the mystery of mirror symmetry.
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