Hodge–Tate and semisimplicity conjecture for hyperplane sections of rational homogeneous spaces

Let $G/P_k$ be a rational homogeneous space, let $r_{G/P_k}$ denote its index, and let $Y$ be a smooth hyperplane section of $G/P_k$. The cohomology $H^*(Y)$ and quantum cohomology $QH^*(Y)$ are considered in the usual sense.

Hodge–Tate and semisimplicity conjecture. The following should hold:

1. $H^*(Y)$ is of Hodge–Tate type if and only if

$$\dim G/P_k<2r_{G/P_k},$$

namely, $G/P_k$ is one of the listed complex Grassmannians, (co)adjoint Grassmannians, or the specified spaces of types $B$, $C$, $D$, $E_6$, and $E_7$. 2. $QH^*(Y)$ is generically semisimple if and only if $G/P_k$ is one of the listed complex Grassmannians, adjoint Grassmannians of type $B$, $C$, $F$, or $G$, or the specified spaces of types $B$, $C$, and $D$.

These statements propose a classification of Hodge–Tate hyperplane sections and of those with generically semisimple quantum cohomology. The source proves the relevant cases for type $A$ and (co)adjoint spaces, while the remaining listed cases are included as part of the conjectural classification; the supplied text gives no resolution status for the full statement.