Ruan–Wang conjecture on quantum cohomology of K-equivalent varieties

Two smooth varieties $X$ and $X'$ are K-equivalent if there exist birational morphisms

$$\phi:Y\to X,\qquad \phi':Y\to X'$$

from a smooth variety $Y$ such that

$$\phi^*K_X={\phi'}^*K_{X'}.$$

Let their quantum cohomology rings be considered over the extended Kähler moduli spaces. Ruan–Wang conjecture. $K$-equivalent smooth varieties have canonically isomorphic quantum cohomology rings over the extended Kähler moduli spaces. This conjecture proposes a quantum-cohomological refinement of the invariance of Betti numbers under $K$-equivalence; the paper studies it through flops and proves invariance in important cases, while the general claim remains open.