Some comments on hyperplane sections of generalised Grassmannians
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 |
Some comments on the table:
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.
The reader of Alexander Kuznetsov: Calabi-Yau and fractional Calabi-Yau categories is left to formulate analogous results for hyperplane sections (and hypersurfaces of low degree) in other generalised Grassmannians. The data from the table will be a good consistency check. These will be partially conjectural (lacking a construction of the required Lefschetz collections), and they might be complicated by the fact that the Lefschetz collections aren't necessarily rectangular. So maybe the Calabi–Yauness mentioned above is only correct for some subcategory.
I fully expect various of these to be known (or conjectured) before. But my memory is not the best, so I guess I need to include an overview of what is known on Grassmannian.info, also known as my external memory.
Update October 23: Sasha Kuznetsov pointed out some issues in the original version of the table. I would like to thank him for this!
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!