Projected gromov-witten varieties in cominuscule spaces

A projected Gromov-Witten variety is the union of all rational curves of fixed degree that meet two opposite Schubert varieties in a homogeneous space X = G/P. When X is cominuscule we prove that the map from a related Gromov-Witten variety is cohomologically trivial. This implies that all (3-point, genus zero) K-theoretic Gromov-Witten invariants of X are determined by projected Gromov-Witten varieties, which extends an earlier result of Knutson, Lam, and Speyer, and provides an alternative version of the ` quantum equals classical' theorem. Our proof uses that any projected Gromov-Witten variety in a cominuscule space is also a projected Richardson variety.