Blow-up formula for quantum cohomology

Yu. I. Manin greatly contributed to several areas of algebraic
geometry, in particular to questions of rationality, and to quantum
cohomology. I will speak about an application of the latter to the
former: the very general cubic fourfold is not rational. The main
ingredient is a blow up formula for genus 0 Gromov-Witten invariants
(the proof is expected soon). Another output is a definition of
"correct" cohomology and GW invariants for singular varieties.