A note on a Griffiths-type ring for complete intersections in Grassmannians

Abstract

We calculate a Griffiths-type ring for smooth complete intersections in Grassmannians. This is the analogue of the classical Jacobian ring for complete intersections in projective space and allows us to explicitly compute their Hodge groups.

Appendix: Fano varieties of K3 type

Recall the following definition (slightly adapted) from [15]:

Definition 5.1

Let X be a smooth projective variety of dimension n, such that $h^{i,0}(X)=0$, $i <n$. Define $h:=\lfloor \frac{n-k}{2}\rfloor$. We say that X is of weak k-Calabi–Yau type if its middle dimensional Hodge structure is numerically similar to a Calabi-Yau k-fold, that is

$$\begin{aligned} h^{n-h,h }=1,\ \ h^{n-h+j,h-j}=0, \ j\ge 1. \end{aligned}$$

We say that X is of strong k-Calabi–Yau type (or simply of k-Calabi–Yau type) if in addition the contraction with any generator $\omega \in H^ {n-h,h }(X)$ induces an isomorphism

$$\begin{aligned} \omega : H^1(T_X) \longrightarrow H^{n-h-1,h+1}(X). \end{aligned}$$

The case of 3-Calabi–Yau is investigated in [15]. We are particularly interested in the 2-Calabi–Yau case, that is, K3 type. Known examples of these varieties in the strong sense include a smooth cubic fourfold $X_3 \subset {\mathbb {P}}^5$, a linear section $Y_1 \subset {{\,\mathrm{Gr}\,}}(3,10)$, cf. [3], and in the weak sense the already mentioned Gushel-Mukai fourfold and the c5- Küchle variety, cf. [18]. Some more examples are found if we allow mild singularities—e.g. cyclic quotient—see [9]. Most of these examples are deeply linked with hyperkähler geometry and derived category problems. Moreover by [19] families of Fano of K3 type (FK3) are likely to be linked with projective families of irreducible holomorphic symplectic manifolds.

These families of FK3 necessarily have to be of dimension greater or equal than four and comparatively high index. This implies we have to apply Theorem 4.4 with caution, since there may be some residual contributions from the ambient space to take into account. However, there is some good news. Denote by $T= {\mathbb {C}}[ {x_I}, {y_i}]$ the basic ambient ring from which we build the Griffiths ring ${\mathcal {U}}$, suitably bigraded as in (7). For a variety X of dimension 2s, a sub-structure of K3-type implies $h^{s+1,s-1}(X)=1$ and $h^{s+t, s-t}(X)=0$, for $t>1$. We therefore look at ${\mathcal {U}}_{i, m}$, with $i \le s-1$ having the above numerological properties. Since the relations in the Griffiths ring ${\mathcal {U}}$ are all in bidegree (0, 1) and (1, 0), m is negative and we have for i in such a range that $T_{i,m}={\mathcal {U}}_{i,m}$. This reduces the problem into a combinatorial one.

Let in fact X be a complete intersection of index m in the Grassmannian ${{\,\mathrm{Gr}\,}}(k,l+k)$ given by the bundle ${\mathcal {F}}= \bigoplus {\mathcal {O}}_G(d_i)$. Denote by $\alpha =c_1({\mathcal {F}})= \sum d_i$. A quick analysis of the polynomial ring T reveals that in order to have

$$\begin{aligned} T_{s-1,m}= {\mathbb {C}}, \ T_{s-t,m}=0 \end{aligned}$$

the weights must be ordered as

$$\begin{aligned} d_1 > d_2 \ge \ldots \ge d_c \end{aligned}$$

and moreover the following equation needs to be satisfied

$$\begin{aligned} 2(k+l-\alpha )=d_1(kl-c-2). \end{aligned}$$

(14)

A computer search confirms that only the already mentioned $X_{2,1} \subset {{\,\mathrm{Gr}\,}}(2,5)$ and $Y_1 \subset {{\,\mathrm{Gr}\,}}(3,10)$ satisfy this relation. They are the well known Gushel-Mukai fourfold and the Debarre-Voisin Fano 20-fold.

However, this does not rule out any other option. Thanks to the residual contributions from the Grassmannian there might be some $X_{d_1, \ldots , d_c}$ with ${\mathcal {U}}_{s-1,m} \ne {\mathbb {C}}$ but still $h^{s-1,s+1}=1$. The condition on the ordering of the weights here might be not required. This is particularly true in the case of linear sections. Indeed, after a first analysis on the cohomology groups of the ambient Grassmannian, we found another example as $X_{1^4} \subset {{\,\mathrm{Gr}\,}}(2,8).$ This is a Fano 8-fold with middle Hodge structure of K3 type. We believe it could lead to a construction of a family of hyperkähler varieties of K3$^{[n]}$ type. We compute its Hodge numbers as

Proposition 5.2

Let $X_{1,1,1,1} \subset {{\,\mathrm{Gr}\,}}(2,8)$ be given by a generic section of ${\mathcal {O}}_G(1)^{\oplus 4}$. The Hodge diamond of $X_{1,1,1,1}$ is

$$\begin{aligned} \begin{array}{ccccccccccccccccc} 0 &{}&{}0 &{}&{}0&{}&{}1 &{}&{} 22 &{}&{}1 &{}&{}0 &{}&{}0 &{}&{}0\\ &{}0 &{}&{} 0 &{}&{} 0 &{}&{} 0&{}&{} 0 &{}&{}0 &{}&{}0 &{}&{} 0&{}\\ &{}&{}0 &{}&{} 0 &{}&{} 0 &{}&{}2 &{}&{}0 &{}&{}0 &{}&{} 0&{}&{}\\ &{}&{}&{}0&{}&{}0 &{} &{} 0 &{} &{}0 &{}&{} 0 &{}&{} 0 &{}&{}&{}\\ &{}&{}&{}&{}0 &{}&{}0 &{}&{} 2 &{}&{}0 &{}&{} 0 &{}&{}&{}&{}\\ &{}&{}&{}&{}&{} 0 &{}&{} 0 &{}&{} 0 &{}&{}0&{}&{}&{} \\ &{}&{}&{}&{}&{}&{}0 &{}&{}1&{}&{}0&{}&{}&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}&{}&{}0&{}&{}0&{}&{}&{}&{}&{}&{}&{}\\ &{}&{}&{}&{}&{}&{}&{}&{}1 &{}&{}&{}&{}&{}&{}&{}&{} \end{array}\end{aligned}$$

with $h^{4,4}_{{{\,\mathrm{van}\,}}}(X)=19$.

Notice that the projective dual of ${{\,\mathrm{Gr}\,}}(2,8)$ is a singular quartic hypersurface in ${\mathbb {P}}^{27}$. Cutting the Grassmannian and the quartic with orthogonal linear subspaces we can link $X_{1,1,1,1} \subset {{\,\mathrm{Gr}\,}}(2,8)$ to a quartic K3 surface $S \subset {\mathbb {P}}^3$. An embedding of the derived category of the quartic K3 inside the derived category of the above linear section is provided in [27, Thm 2.8] . However, we believe that this could be the only exception. Namely, we make the following

Conjecture 5.3

Let $X=X_{d_1, \ldots , d_c} \subset {{\,\mathrm{Gr}\,}}(k,n)$ be a Fano smooth complete intersection of even dimension (that is not a cubic fourfold). Then X is not of K3-type unless

$$\begin{aligned} (\lbrace d_i \rbrace , k,n)=(\lbrace 2,1\rbrace ,2,5), (\lbrace 1,1,1,1\rbrace ,2,8), (\lbrace 1 \rbrace , 3,10). \end{aligned}$$

Our method above can be partially extended to more general vector bundles on other homogeneous varieties. In [8] we analysed a handful more examples and study in details their geometric properties.