Quantum reconstruction for Fano bundles

Abstract

This dissertation discusses Fano vector bundles on projective space and the quantum cohomology of the associated projective bundle. A bundle is said to be Fano if the associated projective bundle is a Fano variety. We prove a theorem that reconstructs the small quantum cohomology ring for the projective bundle from a subset of the quantum multiplication-table. Equivalently, in the spirit of Kontsevich’s first reconstruction theorem, the theorem reconstructs all genus-0 Gromov-Witten invariants from a small set of 2-point invariants. The theorem improves on Kontsevich’s reconstruction theorem, where they both apply, as the required subset of invariants is smaller. Qin and Ruan note that, unlike in the classical case, the quantum cohomology of a projectivised bundle need not be a module over the quantum cohomology of the base. The reconstruction theorem described in this thesis demonstrates that this ‘non-modularity’ contains all the essential quantum information. As an example we discuss the Fano bundle Ω(2)P4 , the second wedge of the cotangent bundle on P4, and calculate the quantum cohomology of the Fano 9-fold given by its projectivisation using the reconstruction theorem. We apply Givental formalism to this result to obtain the quantum periods for some Fano subvarieties embedded in P(Ω(2)P4). These novel calculations are of interest to the Fanosearch programme. In particular, we describe two 4-dimensional Fano subvarieties that have quantum periods suggestive of 4-folds which are not of toric complete-intersection type; they are not contained in a conjecturally complete list of such objects. We observe the presence of the Apery numbers in J-function of the Fano 9-fold P(Ω(2)P4) and comment, after Golyshev, that this may indicate hidden modularity in the J-function.