Today in my arXiv reading I came across Bradley Doyle: Homological Projective Duality for the Plücker embedding of the Grassmannian, and in Section 1.4 there were some comments about the interesting component of a linear section of a Grassmannian that piqued my interest.

I decided to have some fun with more general cases. In type A the following are low-dimensional and possibly of interest:

• $\operatorname{Gr}(4,8)$ does not give a Calabi–Yau category in the hyperplane section, but it can be computed that $\dim\operatorname{HH}_{\pm1}=3$.
• $\operatorname{Gr}(4,9)$ gives a Calabi–Yau category of dimension 3 in the hyperplane section, with the middle cohomology of the hyperplane section having dimensions $1,45,45,1$.

$\operatorname{Gr}(3,11)$ gives another Calabi–Yau category of dimension 3 in the hyperplane section, with the middle cohomology of the hyperplane section having dimensions $1,44,44,1$.

It turns out that the latter two cases are already mentioned in Section 4.5 of Alexander Kuznetsov: Calabi-Yau and fractional Calabi-Yau categories.

For good measure I decided to compute some more middle cohomology of hyperplane sections of orthogonal and symplectic Grassmannians, and see where it could be of interest. Here's a quick overview (hopefully without mistakes now):

$G/P$ middle cohomology $G/P\cap H$ contribution from Lefschetz on $G/P$ comment
$\operatorname{OGr}(2,7)$ $4$ $1$ homological projective dual looks like (a resolution of) triple cover of $\mathbb{P}^{20}$ ramified in the classical projective dual
$\operatorname{OGr}(2,9)$ $6$ $2$ homological projective dual looks like (a resolution of) quadruple cover of $\mathbb{P}^{35}$ ramified in the classical projective dual
$\operatorname{OGr}(3,9)$ $1,47,47,1$ 3-Calabi–Yau category?
$\operatorname{OGr}(2,11)$ $8$ $3$ homological projective dual looks like (a resolution of) quintuple cover of $\mathbb{P}^{54}$ ramified in the classical projective dual
$\operatorname{OGr}(3,11)$ $1,110,1218,1218,110,1$ 4-Calabi–Yau category?
$\operatorname{OGr}(4,11)$ $1,\ldots,1$ 7-Calabi–Yau category?
$\operatorname{SGr}(2,6)$ $4$ $1$ homological projective dual looks like (a resolution of) the triple cover of $\mathbb{P}^{13}$ ramified in the classical projective dual
$\operatorname{LGr}(3,6)$ homological projective dual is noncommutative resolution of quartic hypersurface in $\mathbb{P}^{13}$
$\operatorname{SGr}(2,8)$ $6$ $2$ homological projective dual looks like (a resolution of) the quadruple cover of $\mathbb{P}^{26}$ ramified in the classical projective dual
$\operatorname{SGr}(3,8)$ $2,2$
$\operatorname{LGr}(4,8)$ $3,3$
$\operatorname{SGr}(2,10)$ $8$ $3$ homological projective dual looks like (a resolution of) the quintuple cover of $\mathbb{P}^{43}$ ramified in the classical projective dual
$\operatorname{SGr}(3,10)$ $1,44,44,1$ 3-Calabi–Yau category?
$\operatorname{SGr}(4,10)$ $1,111,1220,1220,111,1$ 5-Calabi–Yau category?
$\operatorname{LGr}(5,10)$ $1,77,354,77,1$ 4-Calabi–Yau category?
$\operatorname{OGr}(2,8)$ $8$ $3$ homological projective dual looks like (a resolution of) the quintuple cover of $\mathbb{P}^{27}$ ramified in the classical projective dual
$\operatorname{OGr}(2,10)$ $10$ $4$ homological projective dual looks like (a resolution of) the sextuple cover of $\mathbb{P}^{44}$ ramified in the classical projective dual
$\operatorname{OGr}(3,10)$ $1,77,358,77,1$ 4-Calabi–Yau category?
$\operatorname{OGr}_+(5,10)$ this variety is homologically and classically self-dual
$\operatorname{OGr}(2,12)$ $12$ $5$ homological projective dual looks like (a resolution of) the septuple cover of $\mathbb{P}^{77}$ ramified in the classical projective dual
$\operatorname{OGr}(3,12)$ $1,\ldots,1$ 6-Calabi–Yau category?
$\operatorname{OGr}(4,12)$ $1,\ldots,1$ Calabi–Yau category?
$\operatorname{OGr}_+(6,12)$ $4$ $2$ homological projective dual looks like (a resolution of the double cover of $\mathbb{P}^{31}$ ramified in the classical projective dual

• By Corollary 4.5 of Alexander Kuznetsov: Calabi-Yau and fractional Calabi-Yau categories we know that the interesting component of $\operatorname{OGr}(2,2n+1)$ is a Calabi–Yau category of dimension 0, so we get that the homological projective dual should be a finite cover of the dual projective space ramified along the classical projective dual.
Update October 25: Enrico Fatighenti pointed out that the $1,44,44,1$ appearing in both the hyperplane sections of $\operatorname{Gr}(3,11)$ and $\operatorname{SGr}(3,10)$ is not a coincidence. Indeed, in Nested varieties of K3 type it is shown that there is a Hodge-theoretic relationship between these hyperplane sections. It is still open to extend this to an equivalence of residual categories, if anyone is up for a challenge!