A Reconstruction Theorem for Quantum Cohomology of Fano Bundles on Projective Space

TL;DR

This paper introduces a reconstruction theorem that derives the quantum cohomology of Fano bundles on projective space from limited Gromov--Witten invariants, with applications to new Fano 9-folds and Fano 3-folds.

Contribution

It provides a novel reconstruction theorem for quantum cohomology of Fano bundles, enabling calculations from minimal Gromov--Witten data and applying it to new geometric examples.

Findings

Reconstructed quantum cohomology of a Fano 9-fold.

Calculated quantum period sequence for a Fano 3-fold.

Demonstrated the theorem's application to the Fanosearch programme.

Abstract

We present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivisation of the bundle from a small number of low-degree Gromov--Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch programme.