Calabi-Yau double coverings of Fano-Enriques threefolds

TL;DR

This paper explores Calabi-Yau threefolds obtained as double coverings of Fano-Enriques threefolds with singularities, calculating their invariants and identifying new examples with Picard number one.

Contribution

It introduces new Calabi-Yau threefolds arising from double coverings of Fano-Enriques threefolds with terminal cyclic quotient singularities.

Findings

Calabi-Yau threefolds are obtained as double coverings of certain Fano-Enriques threefolds.

Calculated invariants for these Calabi-Yau threefolds with Picard number one.

All identified examples are novel in the context of Calabi-Yau geometry.

Abstract

This note is a report on the observation that the Enriques-Fano threefolds with terminal cyclic quotient singularities admit Calabi-Yau threefolds as their double coverings. We calculate the invariants of those Calabi-Yau threefolds when the Picard number is one. It turns out that all of them are new examples.

Tables

Table 1. Table 1. Invariants of Calabi–Yau double coverings

$X_d$	$H_Y^3$	$H_Y\cdot c_2\left(Y\right)$	$h^{1,1}\left(Y\right)$	$h^{1,2}\left(Y\right)$
$X_1$	4	28	1	45
$X_2$	8	32	1	33
$X_3$	2	20	1	37
$X_4$	4	28	1	45

Equations

$$\phi (D)+\phi (\theta (D))$$

$$H^2(X,\mathbb{Z})\simeq \mathrm{Pic}(X),H^2(W,\mathbb{Z})\simeq \mathrm{Pic}(W)$$

$$\xymatrix{X\ar[r]^{\varphi}&W\\ S_{X}\ar[r]^{\varphi|_{S_{X}}}\ar@{{}^{(}->}[u]S\ar@{{}^{(}->}[u]}$$

$$\phi {|}_{S_X}:S_X\to S$$

$$\xymatrix{H^{2}(X,\mathbb{Q})\ar[d]&H^{2}(W,\mathbb{Q})\ar[d]\ar[l]^{\varphi^{*}}\\ H^{2}(S_{X},\mathbb{Q})H^{2}(S,\mathbb{Q})\ar[l]^{{(\varphi|_{S_{X}})}^{*}}}$$

$$H^2(W,\mathbb{Q})\to H^2(S,\mathbb{Q}).$$

$$\xymatrix{H^{2}(X,\mathbb{Z})\ar[d]&H^{2}(W,\mathbb{Z})\ar[d]\ar[l]^{\varphi^{*}}\\ H^{2}(S_{X},\mathbb{Z})_{f}H^{2}(S,\mathbb{Z})_{f}\ar[l]^{{(\varphi|_{S_{X}})}^{*}}}$$

$$(\theta {|}_{S_X}{)}^{\ast }(H_X{|}_{S_X})={\theta }^{\ast }(H_X){|}_{S_X}=H_X{|}_{S_X}$$

$$H^2(S,\mathbb{Z}{)}_f\to H^2(S_X,\mathbb{Z}{)}_f.$$

$$(\phi {|}_{S_X}{)}^{\ast }(H_W{|}_S)={\phi }^{\ast }(H_W){|}_{S_X}=r{H}_X{|}_{S_X}=r{h}^{\prime }$$

$$(\phi {|}_{S_X}{)}^{\ast }:H^2(S,\mathbb{Z}{)}_f\to H^2(S_X,\mathbb{Z}{)}_f$$

$$h^2(Y)\le h^2(X),$$

$$e(Y)=e(X)-24-2(-K_X{)}^3$$

$${\psi }^{\ast }(H_W)\cdot c_2(Y)=(-K_X{)}^3+24,$$

$$\xymatrix{&\widetilde{X}\ar[dr]\ar[dl]\\ Y\ar[dr]^{\psi}X\ar[dl]_{\varphi}\\ W}$$

$$1=\chi (X,{\mathcal{O}}_X)=\frac{1}{24}{c}_2(X)\cdot (-K_X).$$

$$e(S_X)=c_2(X)\cdot (-2K_X)+4(-K_X{)}^3=48+4(-K_X{)}^3.$$

$$e(Y)=2e(W)-e(S)-8=e(X)-\frac{1}{2}e(S_X)=e(X)-24-2(-K_X{)}^3.$$

$$S_Y\cdot c_2(Y)$$

$$=-{\psi }^{\ast }(H_W{)}^3+e(S_Y)$$

$$=-2H_W^3+e(S)$$

$$=-{\phi }^{\ast }(H_W{)}^3+\frac{1}{2}e(S_X)$$

$$=-(-K_X{)}^3+24+2(-K_X{)}^3=(-K_X{)}^3+24.$$

$$H_Y^3=\frac{1}{r^3}(-K_X^3)$$

$$H_Y\cdot c_2(Y)=\frac{1}{r}((-K_X{)}^3+24),$$

$$1\le h^2(W)\le h^2(Y)\le h^2(X)=1,$$

$$(\psi {|}_{S_Y}{)}^{\ast }(H_W{|}_S)={\psi }^{\ast }(H_W){|}_{S_Y}=k(H_Y^{\prime }{|}_{S_Y}).$$

$$H_Y^3={H_Y^{\prime }}^3=\frac{1}{k^3}{\psi }^{\ast }(H_W{)}^3=\frac{2}{k^3}{H}_W^3=\frac{1}{k^3}{\phi }^{\ast }(H_W{)}^3=\frac{1}{r^3{l}^3}(-K_X^3)$$

$$H_Y\cdot c_2(Y)=\frac{1}{k}{\psi }^{\ast }(H_W)\cdot c_2(Y)=\frac{1}{rl}((-K_X^3)+24).$$

$$\chi (Y,H_Y)=\frac{H_Y^3}{6}+\frac{H_Y\cdot c_2(Y)}{12}=\frac{8}{6l^3}+\frac{32}{12l}=\frac{4+8l^2}{3l^3},$$

$$H_Y^3=\frac{1}{r^3}(-K_X^3)$$

$$H_Y\cdot c_2(Y)=\frac{1}{r}((-K_X{)}^3+24).$$

Full text

1. Introduction

The threefolds whose hyperplane sections are Enriques surfaces were studied by G. Fano in a famous paper [5]. The modern proofs for the results of [5] were given in [3]. Such varieties are always singular and their canonical divisors are not Cartier but numerically equivalent to Cartier divisors. We call such threefolds Fano–Enriques threefolds (see Definition 2.1). In this note, we consider Fano–Enriques threefolds whose singularities are terminal cyclic quotient ones. It is worth noting that any Fano–Enriques threefold with terminal singularities admits a ${\mathbb{Q}}\,$-smoothing to one with terminal cyclic quotient singularities ([10]). The canonical coverings (which are double-coverings) of Enriques–Fano threefolds with terminal cyclic quotient singularities are smooth Fano threefolds ([1, 13]). Hence all the singular points of such Enriques–Fano threefolds are of type $\frac{1}{2}(1,1,1)$. Using the classification of smooth Fano threefolds, L.  Bayle ([1]) and T. Sano ([13]) gave a classification of such threefolds. In this note, we observe that all those Enriques–Fano threefolds also admit some Calabi–Yau threefolds as their double covering, branched along some smooth surfaces and those eight singularities. A *Calabi–Yau threefold $Y$ *is a compact Kähler manifold with trivial canonical class such that the intermediate cohomology groups of its structure sheaf are trivial ($h^{1}(Y,\mathcal{O}_{Y})=h^{2}(Y,\mathcal{O}_{Y})=0$). We calculate the invariants of those Calabi–Yau double coverings when their Picard numbers are one (Table 1). It turns out that all those Calabi–Yau threefolds are new examples. Although a number of Calabi–Yau threefolds have been constructed, those with Picard number one are still quite rare. Note that they are primitive and play an important role in the moduli spaces of all Calabi–Yau threefolds ([7]).

2. Calabi–Yau double coverings

As the higher dimensional algebraic geometry being developed, the definition of Fano–Enriques threefolds also has evolved and has been generalized. We adapt the following version of the definition.

Definition 2.1**.**

A three-dimensional normal projective variety $W$ is called a Fano–Enriques threefold if $W$ has canonical singularities, $-K_{W}$ is not a Cartier divisor but numerically equivalent to an ample Cartier divisor $H_{W}$.

Y. Prokhorov proved in [11] that the generic surface in the linear system $|H_{W}|$ is an Enriques surface with canonical singularities and that the Enriques surface is smooth if the singularities of $W$ is isolated and $-K_{W}^{2}\neq 2$. We refer to [2, 6, 12] for more systematic expositions of Fano–Enriques threefolds. In this note, we consider the case that $W$ has only terminal cyclic quotient singularities. We summarize the properties of $W$ ([1, 3, 5, 13]).

1. (1)

All the singularities of $W$ are the type of $\frac{1}{2}(1,1,1)$. 2. (2)

The number of singularities of $W$ is eight. 3. (3)

$-2K_{W}$ is linearly equivalent to $-2H_{W}$. 4. (4)

There is a smooth Fano threefold that covers doubly $W$, branched only at the singularities of $W$.

L. Bayle ([1]) and T. Sano ([13]) gave a classification of smooth Fano threefolds that double cover Fano–Enriques threefolds.

Let $\varphi:X\rightarrow W$ be the double covering, branched along the singularities of $W$. Then $X$ is one of smooth Fano threefolds in Theorem 1.1 in [13]. We want to find a Calabi–Yau threefold that double-covers $W$, using the following theorem which is a special case of Theorem 1.1 in [9].

Theorem 2.2**.**

Let $W$ be a projective three-dimensional variety with singularities of type $\frac{1}{2}(1,1,1)$ such that $h^{1}(W,\mathcal{O}_{W})=h^{2}(W,\mathcal{O}_{W})=0$. Suppose that the linear system $|-2K_{W}|$ contains a smooth surface $S$, then there is Calabi–Yau threefold $Y$ that is a double covering of $W$ with the branch locus $S\cup{\rm{Sing}}(W)$.

Let $p_{1},p_{2}$ be any points of $X$. From the description of those Fano threefolds $X$’s in Theorem 1.1 of [13], one can find an effective divisor $D$ from $|-K_{X}|$ such that $D$ does not contain $p_{1},p_{2}$. Let $\theta$ be the covering involution on $X$, i.e. the quotient $X/\langle\theta\rangle$ is $W$. Therefore, for any point $q\in W$ and we can find an effective divisor $D$ in the linear system $|-K_{X}|$ such that $D\cap\varphi^{-1}(\{q\})=\emptyset$. Note the effective divisor

[TABLE]

belongs to the linear system $|-2K_{W}|$ and it does not contain the point $q$. So the linear system $|-2K_{W}|$ is base-point-free and we can find a smooth surface $S$ from it. Hence, by Theorem 2.2, there is a Calabi–Yau threefold $Y$ that covers doubly $W$, branched along $S$ and singularities of $W$.

Since $H^{i}(X,\mathcal{O}_{X})=0$ and $H^{i}(W,\mathcal{O}_{W})=0$ for $i=1,2$, we have isomorphisms

[TABLE]

by the exponential sequences. Hence we can regard classes of Cartier divisors of $X,W$ as elements of $H^{2}(X,\mathbb{Z}),H^{2}(W,\mathbb{Z})$ respectively. Let $r$ be the index of Fano threefold $X$ (i.e. the largest integer $r$ such that $-K_{X}=rH_{X}$ for some ample divisor $H_{X}$ of $X$).

Now we calculate the invariants of $Y$. For a double covering with dimension higher than two, it is a non-trivial task to calculate the topological invariants even in the case that the base of the covering is smooth. In our case, $S$ is an ample divisor of $W$, so it may be worth trying to apply the Lefschetz hyperplane theorem. However $W$ is not smooth, so the usual Lefschetz hyperplane theorem does not apply here. There are other versions of the Lefschetz hyperplane theorem for singular varieties but they all require that $W-S$ is smooth, which is not true for our case. We prove a type of the Lefschetz hyperplane theorem for $S\subset W$. We say that an element $\alpha$ of an additive Abelian group $G$ is divisible by an integer $k$ if $\alpha=k\alpha^{\prime}$ for some element $\alpha^{\prime}\in G$. $\alpha$ is said to be primitive if it is divisibly by only $\pm 1$. We denote the quotient of $G$ by its torsion part as $G_{f}$.

Lemma 2.3**.**

The map $H^{2}(W,\mathbb{Q})\rightarrow H^{2}(S,\mathbb{Q})$, induced by the inclusion $S\hookrightarrow W$, is injective and the image $H_{W}|_{S}$ in $H^{2}(S,\mathbb{Z})_{f}$ of $H_{W}\in H^{2}(W,\mathbb{Z})$ is divisible by $r$.

Proof.

Consider the commutative diagram:

[TABLE]

where $S_{X}=\varphi^{-1}(S)$ and the vertical maps are inclusions. Note

[TABLE]

is an unramified double covering. We have an induced commutative diagram:

[TABLE]

Note the pull-backs $\varphi^{*}:H^{2}(W,\mathbb{Q})\rightarrow H^{2}(X,\mathbb{Q})$ is injective. Since $S_{X}$ is a smooth ample divisor of $X$, the map $H^{2}(X,\mathbb{Q})\rightarrow H^{2}(S_{X},\mathbb{Q})$ is injective by the Lefschetz hyperplane theorem. So we have the injectivity of the map

[TABLE]

Consider another commutative diagram.

[TABLE]

Note $-K_{X}=rH_{X}$. We note that $\theta^{*}(-K_{X})=-K_{X}$ in $H^{2}(X,\mathbb{Z})$. Since $H^{2}(X,\mathbb{Z})$ has no torsion, $\theta^{*}(H_{X})=H_{X}$ in $H^{2}(X,\mathbb{Z})$ and so

[TABLE]

in $H^{2}(S_{X},\mathbb{Z})_{f}$. Hence $h^{\prime}:=H_{X}|_{S_{X}}$ lies in the image of the map

[TABLE]

Note $\varphi^{*}(H_{W})=-K_{X}=rH_{X}$. Hence

[TABLE]

is divisible by $r$ in $H^{2}(S_{X},\mathbb{Z})_{f}$. Since the map

[TABLE]

is injective, $H_{W}|_{S}$ is divisible by $r$ in $H^{2}(S,\mathbb{Z})_{f}$. ∎

We note that $H_{W}$ is primitive in $H^{2}(W,\mathbb{Z})$. By the above lemma, $H_{W}|_{S}$ is not primitive in $H^{2}(S,\mathbb{Z})_{f}$ when $r>1$. This is different from what the usual Lefschetz hyperplane theorem expects for smooth threefolds.

Proposition 2.4**.**

We have

[TABLE]

[TABLE]

and

[TABLE]

where $e(Y)$ is the topological Euler characteristic of $Y$ and $c_{2}(Y)$ is the second Chern class of $Y$.

Proof.

Consider the following fiber product of two double covers:

[TABLE]

Then it is easy to see that

• (a)

$\widetilde{X}\rightarrow Y$ is an étale double cover and

• (b)

$\widetilde{X}\rightarrow X$ is the double cover branched along a member $\widetilde{S}$ of $|-2K_{X}|$.

Using the fact that $\widetilde{S}$ is an ample divisor of $\widetilde{X}$, one can show $h^{2}(\widetilde{X})=h^{2}(X)$ ([4]). Since $h^{2}(Y)\leq h^{2}(\widetilde{X})$, we have $h^{2}(Y)\leq h^{2}(X).$

Note $e(X)=2e(W)-8$ and $e(S_{X})=2e(S)$. Note $S_{X}\sim-2K_{X}$ and by the Riemann–Roch theorem,

[TABLE]

By the adjunction formula, we have

[TABLE]

So

[TABLE]

Note $\psi^{*}(S)\sim 2S_{Y}$ and $S\sim 2H_{W}$, where $S_{Y}=\psi^{-1}(S)$. So $\psi^{*}(H_{W})\cdot c_{2}(Y)=S_{Y}\cdot c_{2}(Y)$. By the adjunction formula,

[TABLE]

Hence $\psi^{*}(H_{W})\cdot c_{2}(Y)=(-K_{X})^{3}+24$. ∎

We are interested in the case that the Calabi–Yau threefold $Y$ has Picard number one. Hence we assume that $X$ has Picard number one. There are four families of them:

1. $X_{1}$:

complete intersection of a quadric and a quartic in the weighed projective space $\mathbb{P}(1,1,1,1,1,2)$, $r_{1}=1$, $-K_{X_{1}}^{3}=4$, $e(X_{1})=-56$. 2. $X_{2}$:

complete intersection of three quadrics in $\mathbb{P}^{6}$, $r_{2}=1$, $-K_{X_{2}}^{3}=8$, $e(X_{2})=-24$. 3. $X_{3}$:

hypersurface of degree $4$ in $\mathbb{P}(1,1,1,1,2)$, $r_{3}=2$, $-K_{X_{3}}^{3}=16$, $e(X_{3})=-16$. 4. $X_{4}$:

compete intersection of two quadrics in $\mathbb{P}^{5}$, $r_{4}=2$, $-K_{X_{4}}^{3}=32$, $e(X_{4})=0$.

Theorem 2.5**.**

Suppose that $X$ has Picard number one, then $W$, $Y$ have Picard number one,

[TABLE]

and

[TABLE]

where $H_{Y}$ is an ample generator of ${\rm{Pic}}(Y)$.

Proof.

By Proposition 2.4,

[TABLE]

so $W$, $Y$ have Picard number one. Since $Y$ has Picard number one, $\psi^{*}(H_{W})=kH^{\prime}_{Y}$ for some ample generator $H^{\prime}_{Y}$ of ${\rm{Pic}}(Y)$ ( $\simeq H^{2}(Y,\mathbb{Z})$) and a positive integer $k$. Note that $H_{Y}-H_{Y}^{\prime}$ is a torsion element and that $H_{Y}^{\prime}$ is primitive in $H^{2}(Y,\mathbb{Z})$. We also note that $S_{Y}$ is a smooth ample divisor of $Y$. By the Lefschetz hyperplane theorem, $H^{\prime}_{Y}|_{S_{Y}}$ is primitive in $H^{2}(S_{Y},\mathbb{Z})_{f}$. By Lemma 2.3, $H_{W}|_{S}$ is divisible by $r$ in $H^{2}(S,\mathbb{Z})_{f}$. So its image $(\psi|_{S_{Y}})^{*}(H_{W}|_{S})$ in $H^{2}(S_{Y},\mathbb{Z})_{f}$ is divisible by $r$. Note

[TABLE]

So $k$ is divisible by $r$. Let $k=lr$ for some positive integer $l$. We will show that $l=1$. Note

[TABLE]

and

[TABLE]

For $X_{1},X_{3},X_{4}$, the condition of $H_{Y}^{3}$ being a positive integer requires that $l=1$. For $X_{2}$, by the Riemman–Roch theorem, we have

[TABLE]

which should be an integer. So we have $l=1$ also in this case. Therefore,

[TABLE]

and

[TABLE]

∎

By Proposition 2.4 and the relation $e(Y)=2(h^{1,1}(Y)-h^{1,2}(Y))$, we can determine all the Hodge numbers of $Y$. We list the invariants of the Calabi–Yau threefolds $Y$’s in Table 1. It turns out that they are all new examples . See Appendix I of [8] for a list of known examples of Calabi–Yau threefolds of Picard number one.

Note that the invariants of $Y_{1}$ and those of $Y_{4}$ overlap. Consider the commutative diagram in the proof of Proposition 2.4. For $X_{1}$, the branch locus of $\widetilde{X}_{1}\rightarrow X_{1}$ is a quadric section, thus $\widetilde{X}_{1}$ is a $(2,2,4)$-weighted complete intersection of $\mathbb{P}(1,1,1,1,1,1,2)$. For $X_{4}$, the branch locus of $\widetilde{X}_{4}\rightarrow X_{4}$ is a quartic section, thus $\widetilde{X}_{4}$ is also a $(2,2,4)$-weighted complete intersection of $\mathbb{P}(1,1,1,1,1,1,2)$. Therefore, $\widetilde{X}_{1}$ and $\widetilde{X}_{4}$ are in the same family. Since $Y_{1}$ and $Y_{4}$ are étale $\mathbb{Z}_{2}$-quotients of $\widetilde{X}_{1}$ and $\widetilde{X}_{4}$ respectively, they have the same invariants.