Hodge number of Fano varieties

Question

Let $k=\mathbb{C}$ be the base field. Let $X_g$ be a smooth prime Fano fourfold of index $2$ and genus $g=7,8,9$. What are the values of $h^{1,3}(X_g)$?

I semi-convinced myself that $h^{1,3}(X_9) = 0$ as follows, using a sarkisov link constructed by Prokhorov and Zaidenberg [3,Theorem 2.1] $b_4(X_9) = 4$, moreover the fourth Chow group is a free abelian group of rank $4$ [1,Proposition 3.7]. However, the relationship between the Chow ring and cohomology ring is in general not straightforward so it not a proof as far as I can tell.

Disclaimer: I am aware that the computation is possible using the methods of Kuechle [2,Proposition 4.7] (all the $X_{g}$ are zero sections of explicitly known vector bundles on Grassmanians) and started undertaking this. However since the computations involved are long and complicated, I would like to check with some experts that these invariants are not contained in the literature before continuing. Also looking at the literature it seems that the varieties involved are more "special" than typical zero sets of sections of homogeneous vector bundles over Grassmannians, so perhaps there is a simpler way...

References.

Geometry of the genus 9 Fano 4-folds Han, Frédéric. Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1401-1434.
On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians Oliver Kuechle .Math. Z. 218, 563-575 (1995).
New examples of cylindrical Fano fourfolds. Yuri Prokhorov and Mikhail Zaidenberg. Advanced Studies in Pure Mathematics 75, 2017 Algebraic Varieties and Automorphism Groups pp. 443–463,

Do you mean linear sections of $\mathrm{OGr}_+(5,10)$, $\mathrm{Gr}(2,6)$, and $\mathrm{LGr}(3,6)$? — Sasha
– Sasha, Commented 12 hours ago
As far as I understand, In the paper arxiv.org/abs/1406.4891 there are descriptions of all three in terms of ordinary Grassmannians for all three (, in $Gr(2,5), Gr(2,6), Gr(3,6)$ all of them are stated on page 4. — Nick L
– Nick L, Commented 12 hours ago
Sorry, I may have misused the phrase "linear section", to mean zero set of homogeneous vector bundle... — Nick L
– Nick L, Commented 12 hours ago

Sasha · Accepted Answer · 2025-09-03 15:41:30Z

All these fourfolds have $h^{p,q} = 0$ for $p \ne q$. One way to see this is by noting that all of them admit a full exceptional collection, hence their Hochschild homology $\mathrm{HH}_i = 0$ for $i \ne 0$ (by Hochschild--Kostant--Rosenberg this gives the vanishing of $h^{p,q}$ for $p \ne q$).

To prove the existence of a full exceptional collection one can use Homological Projective Duality. It follows from it that the orthogonal complement to the part of the derived category restricted from the ambient Grassmannian is

the derived category of 12 points, for $g = 7$;
the derived category of a smooth cubic surface, for $g = 8$;
the derived category of 4 points, for $g = 9$. Since all these categories have a full exceptional collection, the same is true for the derived categories of the corresponding fourfolds.

By the way, the same approach and argument also works for $g = 10$, where the orthogonal complement is the derived category of 2 points.

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