Dubrovin's reduced quantum cohomology conjecture

Let $X$ be a variety, let $H^{p,p}(X)$ denote its Hodge $(p,p)$-cohomology, and let $M'$ be the Frobenius submanifold associated with the subspace

$$\bigoplus_p H^{p,p}(X) \subset H^*(X).$$

The reduced quantum cohomology of $X$ is the quantum cohomology corresponding to $M'$, and an exceptional collection in $D^b(X)$ is a sequence of exceptional objects with the required semiorthogonality.

Dubrovin's reduced quantum cohomology conjecture. The variety $X$ has generically semisimple reduced quantum cohomology, that is, $M'$ is generically semisimple, if and only if there exists an exceptional collection of length

$$\operatorname{rk} \bigoplus_p H^{p,p}(X)$$

in $D^b(X)$.

This modification accounts for varieties whose cohomology is not concentrated in Hodge bidegrees $(p,p)$, since such varieties cannot have an exceptional system of the length required by the full conjecture. The paper proves stability of generic semisimplicity for $(p,p)$-quantum cohomology under blow-ups at points, but does not resolve this modified conjecture.