On the atomic decomposition from quantum multiplication and its applications

In a talk given at HSE in 2019, I proposed a way to apply genus zero Gromov-Witten invariants to questions of rationality of algebraic varieties. The main idea is based on a loose analogy with the morsification of an isolated critical point of a holomorphic function. The cohomology of any smooth projective variety defined over a field of characteristic zero (not necessarily algebraically closed) splits non-canonically into the sum of so-called atoms, whose isomorphism classes as noncommutative motives are canonical. Recently Hiroshi Iritani proved so-called blowup formula for quantum cohomology, which is the key result for the program, and describes the structure of atoms for the blowup. Based on Iritani's result one can conclude now e.g. that the very general cubic 4-fold is not rational. I'll give a review of new invariants, and propose several conjectures refining the structure of atoms and related to Bridgeland stability structures on derived categories of coherent sheaves. The talk is based on the ongoing project with L. Katzarkov, T. Pantev and T. Yu, as well as a related another project with D. Auroux and L. Katzarkov.