Manifolds of low dimension with trivial canonical bundle in Grassmannians

Abstract

We study fourfolds with trivial canonical bundle which are zero loci of sections of homogeneous, completely reducible bundles over ordinary and classical complex Grassmannians. We prove that the only hyper-Kähler fourfolds among them are the example of Beauville and Donagi, and the example of Debarre and Voisin. In doing so, we give a complete classification of those varieties. We include also the analogous classification for surfaces and threefolds.

Appendices

Appendix A: Euler characteristic

The computation of the Euler characteristic of the trivial bundle ${{{\mathcal {O}}}}_Y$ can be done in two different ways. The first one is applicable in general only for the symplectic and odd orthogonal Grassmannians, while the second one can be applied to all the Grassmannians (and actually is lighter in terms of computational time).

We explain the first method in Sect. A.1. For the second method, we use the fact that the cohomology ring of the classical Grassmannians is well known. In Sect. A.2.1 we recall Borel’s presentation of the cohomology ring of orthogonal Grassmannians, and in Sect. A.2.2 we explain how to integrate products of Chern classes of homogeneous vector bundles.

1.1 A.1 Computation in the symplectic and odd orthogonal case

For symplectic and odd orthogonal Grassmannians one knows that one can choose multiplicative generators of the cohomology to be the Chern classes of the quotient bundle ${{{\mathcal {Q}}}}$. This is the tautological quotient bundle of the classical Grassmannian in which the symplectic and odd orthogonal Grassmannians embed naturally (as a reference, one can see [19, Theorem 8, Theorem 12]). For instance, ${{\mathrm{IGr}}}(k, 2n)$ embeds in ${{\mathrm{Gr}}}(k, 2n)$ as the zero locus of a section of $\Lambda ^2{{{\mathcal {U}}}}^*$. Then, suppose to be able to express the Chern classes of the bundle ${{{\mathcal {F}}}}$ which defines Y in ${{\mathrm{IGr}}}(k, 2n)$ in terms of the Chern classes of ${{{\mathcal {Q}}}}$, which live in the cohomology of ${{\mathrm{Gr}}}(k, 2n)$. Then, the computation can be made in this last space. Indeed, $[ Y] = {{\mathrm{c_{top}}}}({{{\mathcal {F}}}})$ in the cohomology ring of ${{\mathrm{IGr}}}$, $[ {{\mathrm{IGr}}}] ={{\mathrm{c_{top}}}}(\Lambda ^2{{{\mathcal {U}}}}^*)$ in the cohomology ring of ${{\mathrm{Gr}}}$, and

$$\begin{aligned} \int _{Y}(\cdot )=\int _{{{\mathrm{IGr}}}}(\cdot ) [Y] =\int _{{{\mathrm{Gr}}}}(\cdot )[ Y] [ {{\mathrm{IGr}}}] . \end{aligned}$$

Therefore, one has:

$$\begin{aligned} \chi ({{{\mathcal {O}}}}_Y)= & {} \int _Y {{\mathrm{td}}}(T_Y)=\int _{{{\mathrm{IGr}}}} \frac{{{\mathrm{td}}}(T_{{{\mathrm{IGr}}}})}{{{\mathrm{td}}}({{{\mathcal {F}}}})} {{\mathrm{c_{top}}}}({{{\mathcal {F}}}})\\= & {} \int _{{{\mathrm{Gr}}}}\frac{{{\mathrm{td}}}(T_{{{\mathrm{Gr}}}})}{{{\mathrm{td}}}({{{\mathcal {F}}}})}\frac{{{\mathrm{c_{top}}}}(\Lambda ^2({{{\mathcal {U}}}}^*))}{{{\mathrm{td}}}(\Lambda ^2({{{\mathcal {U}}}}^*))} {{\mathrm{c_{top}}}}({{{\mathcal {F}}}}). \end{aligned}$$

In the second equality we have used the fact that ${{\mathrm{td}}}$ is multiplicative with respect to short exact sequences, and we have applied this to the normal sequence for Y:

$$\begin{aligned} 0\rightarrow T_Y \rightarrow T_{{{\mathrm{IGr}}}}|_Y \rightarrow {{{\mathcal {F}}}}|_Y \rightarrow 0 \end{aligned}$$

Then, as in the case of the classification of fourfolds in the classical Grassmannian, one can use the MACAULAY2 package SCHUBERT2 to do the computation. In the symplectic Grassmannian concretely there is essentially one bundle for which one needs to find the relation with the Chern classes of ${{{\mathcal {Q}}}}$, namely $({{{\mathcal {U}}}}^{\perp }/{{{\mathcal {U}}}})(1)$. This is given by the exact sequence:

$$\begin{aligned} 0\rightarrow {{{\mathcal {U}}}}^{\perp }/{{{\mathcal {U}}}}(1) \rightarrow {{{\mathcal {Q}}}}(1) \rightarrow {{{\mathcal {U}}}}^*(1) \rightarrow 0 \end{aligned}$$

For the orthogonal Grassmannian one has to consider also the bundles which correspond to half integer sequences, and in particular the spin bundles. By relating the exterior and symmetric powers of these bundles with the “classical” bundles in the different cases, we can do the computation. For example, for ${{\mathrm{OGr}}}(n, 2n+1)$, the spin bundle is just the “square root” of ${{{\mathcal {O}}}}(1)$, so its first Chern class is half the one of the ample line bundle ${{{\mathcal {O}}}}(1)$.

1.2 A.2 Cohomology of the even orthogonal Grassmannian

The method explained earlier cannot be used in general for the even orthogonal Grassmannian, because its cohomology is a little bit more complicated. In this case in fact, the Chern classes of the tautological quotient bundle ${{{\mathcal {Q}}}}$ do not generate multiplicatively the cohomology ring. One way to proceed is to use a “good” presentation of the cohomology, for which it is relatively easy to understand what the Chern classes of the homogeneous bundles are, and so to compute the integral in the equation mentioned above. The picture we are going to present applies actually, with appropriate modifications, to the other cases too.

1.2.1 A.2.1 Borel’s presentation of the cohomology ring

We present the multiplicative structure of the cohomology ring of the even orthogonal Grassmannian; it is usually referred to as Borel’s presentation of the cohomology (as a reference see [7], where the more general case of a homogeneous variety is treated). The idea is to inject the cohomology ring into one which is better understood, namely that of a complete flag variety. This has the property that every irreducible homogeneous bundle has rank one. Therefore it will always be possible to split completely a not necessarily irreducible bundle and to compute its Chern class as the product of the Chern classes of the line bundles of the splitting.

To be more precise, one considers the projection map $\pi :{{\mathrm{SO}}}(2n)/B \rightarrow {{\mathrm{SO}}}(2n)/P_k={{\mathrm{OGr}}}(k, 2n)$, where B is a Borel subgroup contained in $P_k$. The homogeneous space ${{\mathrm{SO}}}(2n)/B$ is the complete flag $OF(1,\dots ,n, {\mathbb {C}}^{2n})$ of isotropic planes in ${\mathbb {C}}^{2n}$. The fiber of $\pi$ over a point $[R]\in {{\mathrm{OGr}}}(k, 2n)$ is isomorphic to ${{\mathrm{F}}}(1,\ldots ,k, R)\times OF(1,\dots ,n-k, R^{\perp }/R)$, where the first factor is the usual complete flag in R. As $\pi$ is a fibration, the pull-back morphism $\pi ^*$ is an injection. Therefore, after pulling back, one can work in the cohomology of the flag variety G / B, where $G={{\mathrm{SO}}}(2n)$.

What one gains, as already anticipated, is the fact that this cohomology is well known and every homogeneous vector bundle can be split as the sum of line bundles. Indeed, let us denote by X(T) the characters of the maximal torus T in B. Then one has a morphism (defined in [7, Section 3]):

$$\begin{aligned} c: S_{{\mathbb {Q}}}[X(T)]\rightarrow {{\mathrm{H}}}^*_{{\mathbb {Q}}}(G/B) \end{aligned}$$

from the symmetric algebra on the characters with rational coefficients to the rational cohomology of G / B. This morphism is surjective, and so identifies a quotient $S_{{\mathbb {Q}}}[X(T)]/I$ with ${{\mathrm{H}}}^*_{{\mathbb {Q}}}(G/B)$. The ideal I can be computed explicitly as the ideal generated by invariant polynomials (without constant terms) under the natural action of the Weyl group W of G on $S_{{\mathbb {Q}}}[X(T)]$. On the other hand, we will need to be able to write explicitly the morphism c. One has:

$$\begin{aligned} c(f)=\sum _{w\in W|l(w)=deg(f)}\Delta _w(f) X^w \end{aligned}$$

for f homogeneous in $S_{{\mathbb {Q}}}[X(T)]$, where $X^w$ is the Schubert cohomology class corresponding to the Weyl element w (in the usual Schubert presentation of the cohomology ring of homogeneous spaces). Moreover, given a reduced decomposition of $w=s_{i_1} \dots s_{i_{l(w)}}$ in terms of simple reflections, $\Delta _w=\Delta _{s_{i_1}}\circ \dots \circ \Delta _{s_{i_{l(w)}}}$, where

$$\begin{aligned} \Delta _{s_i}(f)=\frac{f-s_i(f)}{\alpha _i}, \end{aligned}$$

$\alpha _i$ being the i-th simple root. The value of $\Delta _w(f)$ doesn’t depend on the chosen reduced decomposition (again, refer to [7, Theorem 1]).

1.2.2 A.2.2 Chern class of homogeneous bundles

Now, having this in mind, the last step to do the computation of the Euler characteristic is to express the Chern classes of a homogeneous vector bundle on $G/P_k$ in ${{\mathrm{H}}}^*(G/B)$. As already pointed out, a homogeneous completely reducible bundle splits in ${{\mathrm{H}}}^*(G/B)$ as the sum of line bundles. These line bundles correspond to representations of B, as explained in Sect. 2.1, i.e. to elements of X(T). Fix ${{{\mathcal {F}}}}$ on $G/P_k$ coming from a representation V of $P_k$. Then one has the weight space decomposition $V=\oplus _{\mu \in X(T)}V_{\mu }^{m_{\mu }}$. As a consequence,

$$\begin{aligned} \pi ^*({{{\mathcal {F}}}})\sim \oplus _{\mu \in X(T)}{{{\mathcal {L}}}}_{\mu }^{m_{\mu }} \end{aligned}$$

where ${{{\mathcal {L}}}}_{\mu }$ is the line bundle corresponding to $\mu \in X(T)$. Here, the symbol $\sim$ stands for “are the same as T-homogeneous bundles”, which implies that they have the same Chern classes. The last ingredient is the fact that the Chern class of ${{{\mathcal {L}}}}_{\mu }$ is represented inside $S_{{\mathbb {Q}}}[X(T)]$ by $1+\mu$. As a consequence

$$\begin{aligned} {{\mathrm{Chern}}}(\pi ^*({{{\mathcal {F}}}}))=\prod _{\mu \in X(T)}(1+\mu )^{m_{\mu }} \end{aligned}$$

Knowing this, we can compute the Chern class of the bundle, products of cohomology classes, integrations, etc. In particular, the integration on $G/P_k$ of a class f of the right degree is given by computing $\Delta _{w_0}(f)$, where $w_0$ is the longest element in $W/W(P_k)$.

Remark A.1

(The case (ob5), Table 4)  It is the only case in which the Euler characteristic is not equal to 2, but to 4. In order to understand better why this happens, we studied in more detail the cohomology of ${{{\mathcal {O}}}}_Y$ by using the Koszul complex associated to the bundle ${{{\mathcal {F}}}}$. The method is standard (see Sect. 3.2), the only difficulty in this case is to express the bundle $\Lambda ^k{{{\mathcal {F}}}}^*$ as a sum of irreducible homogeneous bundles, but this can be done using the program LiE [20]. What one finds is that the variety (ob5) is not connected, and actually it consists of two connected components, which are therefore Calabi–Yau varieties. Two questions which can be asked is whether these two components are isomorphic, and if there is a more geometric explanation for the existence of these two components, as is the case for the variety of zeroes of $S^2{{{\mathcal {Q}}}}$ in ${{\mathrm{Gr}}}(m,2m)$.

Appendix B: Tables

In this appendix we report all tables of the varieties we have found, as indicated in the classification theorems in the paper. The labelling of the varieties follows essentially the subdivision of the classification’s proofs in lemmas and propositions. In the following we explicit some rules we adopted.

For the ordinary (respectively symplectic, orthogonal) Grassmannians ${{\mathrm{Gr}}}(k,n)$ (${{\mathrm{IGr}}}(k,m)$, ${{\mathrm{OGr}}}(k,m)$) with $k\le n$, if $k=2$ then the varieties are labelled by the letter (b) (resp. (sb), (ob)), if $k=3$ they are labelled by (c) (resp. (sc), (oc)), and if $k\ge 4$, labels start with the letter (d) (resp. (sd), (od)).

Some special cases are to be taken into account. For the odd orthogonal Grassmannians, varieties inside ${{\mathrm{OGr}}}(k-2,2k+1)$ are labelled by (ox) and varieties inside ${{\mathrm{OGr}}}(k,2k+1)$ by (oy). For the even orthogonal Grassmannians, varieties inside ${{\mathrm{OGr}}}(k-2,2k)$ are labelled by (ow) and varieties inside ${{\mathrm{OGr}}}(k,2k)$ by (oz). Finally, varieties inside ${{\mathrm{OGr}}}(k-1,2k)$ are labelled by the letter (oe).

Table 1 Fourfolds in ordinary Grassmannians, see Theorem 3.1

Table 2 Fourfolds in symplectic Grassmannians, see Theorem 4.1

Table 3 Fourfolds in odd orthogonal Grassmannians, see Theorem 4.1

Table 4 Fourfolds in even orthogonal Grassmannians, see Theorem 4.1

Table 5 Fourfolds in ${{\mathrm{OGr}}}(n-1,2n)$, see Theorem 4.21

Table 6 Threefolds in ordinary Grassmannians, see Theorem 5.1

Table 7 Threefolds in symplectic Grassmannians, see Theorem 5.2

Table 8 Threefolds in odd orthogonal Grassmannians, see Theorem 5.2

Table 9 Threefolds in even orthogonal Grassmannians, see Theorem 5.2

Table 10 Threefolds in ${{\mathrm{OGr}}}(n-1,2n)$, see Theorem 5.3

Table 11 Surfaces in ordinary Grassmannians, see Theorem 5.4

Table 12 Surfaces in symplectic Grassmannians, see Theorem 5.5

Table 13 Surfaces in odd orthogonal Grassmannians, see Theorem 5.5

Table 14 Surfaces in even orthogonal Grassmannians, see Theorem 5.5

Table 15 Surfaces in ${{\mathrm{OGr}}}(n-1,2n)$, see Theorem 5.6

In the tables concerning K3 surfaces, we have marked the varieties which have already been studied by Mukai with M. In particular: cases M.(b10), M.(oy5), M.(b13), M.(c6) are in [14]; case M.(c9) is in [16]; cases M.(ox5), M.(d3) are in [15]. There are many other cases which have not been examined yet, and they are worth being considered in more detail, as we intend to do in the next future.