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A Global Torelli Theorem for Certain Calabi-Yau Threefolds

Abstract

We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of \({{\mathbb {P}}}^3\) branched along six stable hyperplanes.

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References

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Acknowledgements

We would like to thank Chin-Lung Wang for his interest in our work, especially posing the classification question based on his own work [21] on the WP metric completion program of Calabi-Yau moduli spaces. We are grateful to several anonymous referees for the valuable comments that help us improve the quality and readability of the paper. The first author started his research in Hodge theory by learning the global Torelli theorem for K3 surfaces, together with Yi Zhang when both of us were graduate students at the Chinese University of Hong Kong. His sudden departure leaves us with a great grief. We would like to devote this work to the memory of Yi Zhang.

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Correspondence to Jinxing Xu.

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To the memory of Yi Zhang.

This work is supported by the National Key R and D Project (2020YFA0713100), the Fundamental Research Funds for the Central Universities (No. WK3470000018), Youth Innovation Promotion Association CAS, National Natural Science Foundation of China (Grant No. 11721101), Anhui Provincial Natural Science Foundation (2008085MA04) and Anhui Initiative in Quantum Information Technologies (AHY150200).

Comparison with Rohde’s Example

Comparison with Rohde’s Example

Given distinct \(a,b,c\in {{\mathbb {C}}}\backslash \{0,1\}\), Rohde [14] constructed a singular Calabi-Yau threefold \(X^{'}\) in the following way:

Let W be the surface in the weighted projective space \({{\mathbb {P}}}(2,2,1,1)\) defined by the equation \(y_1^3+y_2^3+x_0x_1(x_1-x_0)(x_1-ax_0)(x_1-bx_0)(x_1-cx_0)=0\). Let F be the Fermat curve in \({{\mathbb {P}}}^2\) defined by the homogeneous equation \(z_0^3+z_1^3+z_2^3=0\). Then the cyclic group \(G={{\mathbb {Z}}}/3{{\mathbb {Z}}}\) acts on W and F. Fixing a generator \(\sigma \) of G and let \(\omega \) be a fixed primitive cubic root of unity. We define these actions explicitly:

$$\begin{aligned} \begin{aligned} \sigma : W&\rightarrow W, \\ [x_0:x_1:y_1:y_2]&\mapsto [x_0:x_1:\omega y_1:\omega y_2],\\ \sigma :&F\rightarrow F,\\ [z_0:z_1:z_2]&\mapsto [\omega z_0:z_1:z_2]. \end{aligned} \end{aligned}$$

Rohde [14] constructed the Calabi-Yau threefold as a crepant resolution of the quotient threefold \(X^{'}:=W\times F/G\), where G acts on \(W\times F\) in the diagonal way. Moreover, varying the parameters abc in \({{\mathbb {C}}}\backslash \{0,1\}\), Rohde obtains a family of Calabi-Yau threefolds \({\mathcal {X}}^{'}\rightarrow {\mathfrak {M}}_{1,6}\), where we recall that \({\mathfrak {M}}_{1,6}\) is the moduli space of ordered six distinct points in \({{\mathbb {P}}}^1\). The main goal of this section is to show Rohde’s family is birationally equivalent to the family \({\mathcal {X}}_{AR}\rightarrow {\mathfrak {M}}_{3, 6}\), which is the universal family of cyclic triple covers of \({{\mathbb {P}}}^3\) branched along six hyperplanes in general position.

We first analyze the structure of the singular surface W. In general, if \(X_1\) and \(X_2\) are two varieties with G-action, we say \(X_1\) and \(X_2\) are G-birationally equivalent if there exists a birational map from \(X_1\) to \(X_2\) compatible with the G-action.

Let C be the cyclic triple cover of \({{\mathbb {P}}}^1\) branched along the six points \(\{0, 1, \infty , a, b, c\}\) whose affine model is the curve in \({{\mathbb {C}}}^2\) defined by the equation \(y^3-x(x-1)(x-a)(x-b)(x-c)=0\). Then G acts on C in the following way:

$$\begin{aligned} \begin{aligned} \sigma : C&\rightarrow C,\\ (x,y)&\mapsto (x, \omega y). \end{aligned} \end{aligned}$$

Let G act on the product \(C\times F\) diagonally, then G acts on the quotient \(C\times F/ G\) through the G-action on the first factor C.

Lemma A.1

W is G-birationally equivalent to the quotient \(C\times F/ G\).

Proof

Let \(F_0\) be the affine curve \(\{(z_1,z_2)\in {{\mathbb {C}}}^2| 1+z_1^3+z_2^3=0\}\), and define a G-action on \(F_0\) by \(\sigma (z_1,z_2)=(\omega z_1,\omega z_2)\). Then \(F_0\) is G-birational to the Fermat curve F. Define the following morphism:

$$\begin{aligned} \begin{aligned} C\times F_0&\rightarrow W, \\ (x,y,z_1,z_2)&\mapsto (x, z_1 y, z_2 y). \end{aligned} \end{aligned}$$

It is easy to see this morphism induces a G-birational equivalent between \(C\times F_0/ G\) and W. So W is G-birationally equivalent to \(C\times F/ G\). \(\square \)

Now we consider the following six hyperplanes in \({{\mathbb {P}}}^3\) which are in general position

$$\begin{aligned} H_i: X_i=0 \ (0\le i\le 3), \ \ H_4: \sum _{i=0}^3X_i=0, \ \ H_5: X_0+a X_1+b X_2+c X_3=0, \end{aligned}$$

where \([X_0:\cdots :X_3]\) is the homogeneous coordinates on \({{\mathbb {P}}}^3\). Let X be the cyclic triple cover of \({{\mathbb {P}}}^3\) branched along \(\sum _{i=0}^5 H_i\).

In order to analyze the structure of X, we define some auxiliary varieties.

Let \(u_1\), \(v_1\) be linear functions of u, v defined by the following relations:

$$\begin{aligned} \left\{ \begin{array}{ll} u_1 &{} =1+u+v, \\ v_1 &{} =a+bu+cv. \end{array} \right. \end{aligned}$$
(A.1)

We define Y as the following affine variety:

$$\begin{aligned} Y=\left\{ (t_1,u,v,y_1)\in {{\mathbb {C}}}^4| y_1^3=\frac{uvt_1(t_1+1)}{u_1v_1(v_1-u_1)}\right\} . \end{aligned}$$

Let S be the following affine surface:

$$\begin{aligned} S=\Bigg \{(w, u, v)\in {{\mathbb {C}}}^3| w^3=\frac{uv}{u_1v_1(v_1-u_1)}\Bigg \}. \end{aligned}$$

Let G acts on S by \(\sigma (w, u, v)=(\omega ^2 w, u, v)\).

Lemma A.2

We have the following birational isomorphsims:

  1. (1)

    X is birationally equivalent to Y.

  2. (2)

    Y is birationally equivalent to \(S\times F /G\), where G acts on \(S\times F\) diagonally.

  3. (3)

    S is G-birationally equivalent to \(C\times F /G\).

Proof

(1) We take the following affine model of X:

$$\begin{aligned} X_1=\{(x_1,x_2,x_3,y)\in {{\mathbb {C}}}^4| y^3=x_1x_2x_3(1+x_1+x_2+x_3)(1+ax_1+bx_2+cx_3)\}. \end{aligned}$$

Under the coordinate transformation

$$\begin{aligned} \left\{ \begin{array}{ll} x_1 &{} =t, \\ x_2 &{} =tu,\\ x_3 &{} =tv,\\ y &{} =y. \end{array} \right. \end{aligned}$$

the hypersurface \(X_1\) is birationally equivalent to the following hypersurface in \({{\mathbb {C}}}^4\):

$$\begin{aligned} X_2=\{(t,u,v,y)\in {{\mathbb {C}}}^4| y^3=t^3uv(1+u_1t)(1+v_1t)\}, \end{aligned}$$

where \(u_1\), \(v_1\) are defined by the equations (A.1). Then we can see \(X_2\) is birational to Y under the following coordinate transformation:

$$\begin{aligned} \left\{ \begin{array}{ll} u &{} =u, \\ v &{} =v,\\ t &{} =(\frac{1}{v_1}-\frac{1}{u_1})t_1-\frac{1}{u_1},\\ y &{} =tu_1v_1(\frac{1}{u_1}-\frac{1}{v_1})y_1. \end{array} \right. \end{aligned}$$

(2) It is direct to see the following affine curve \(F_1\) is G-birationally equivalent to the Fermat curve F:

$$\begin{aligned} F_1=\{(t_1,x)\in {{\mathbb {C}}}^2| x^3=t_1(t_1+1)\}, \end{aligned}$$

where G acts on \(F_1\) by \(\sigma (t_1, x)=(t_1, \omega x)\).

Then the following rational map induces the desired G-birationally equivalence between \(S\times F /G\) and Y:

$$\begin{aligned} \begin{aligned} S\times F_1&\rightarrow Y, \\ (w, u, v, t_1, x)&\mapsto (t_1, u, v, wx). \end{aligned} \end{aligned}$$

(3) From the equations (A.1), we can view \((w, u_1, v_1)\) as a coordinate system on \({{\mathbb {C}}}^3\), and we make the following coordinate transformation:

$$\begin{aligned} \left\{ \begin{array}{ll} w &{} =w, \\ u_1 &{} =t_2,\\ v_1 &{} =t_2z. \end{array} \right. \end{aligned}$$

Under this coordinate transformation, we see S is G-birationally equivalent to the following surface:

$$\begin{aligned} S_1=\{(w,t_2,z)\in {{\mathbb {C}}}^3| w^3=\frac{(A_1t_2+a_1)(B_1t_2+b_1)}{t_2^3z(z-1)}\}, \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{array}{ll} A_1 &{} =\frac{b-z}{b-c}, \\ B_1 &{} =\frac{c-z}{c-b},\\ a_1 &{} =\frac{a-b}{b-c},\\ b_1 &{} =\frac{a-c}{c-b}, \end{array} \right. \end{aligned}$$

and G acts on \(S_1\) by \(\sigma (w, t_2, z)=(\omega ^2 w, t_2, z)\).

A further coordinate transformation:

$$\begin{aligned} \left\{ \begin{array}{ll} z&{}= z,\\ t_3 &{} =\frac{t_2+\frac{a_1}{A_1}}{\frac{b_1}{B_1}-\frac{a_1}{A_1}}, \\ w_1 &{} =\frac{wt_2}{A_1B_1(\frac{b_1}{B_1}-\frac{a_1}{A_1})}, \end{array} \right. \end{aligned}$$

shows that \(S_1\) is G-birational to the surface:

$$\begin{aligned} S_2=\{(z,t_3,w_1)\in {{\mathbb {C}}}^3|w_1^3=\frac{t_3(t_3+1)}{z(z-1)A_1B_1(A_1b_1-B_1a_1)} \}, \end{aligned}$$

where G acts on \(S_2\) by \(\sigma (z, t_3, w_1)=(z, t_3, \omega ^2 w_1)\).

Now let \(C_1\) be the following affine curve:

$$\begin{aligned} C_1:=\{(z,w_2)\in {{\mathbb {C}}}^2| w_2^3=z(z-1)A_1B_1(A_1b_1-B_1a_1)\}, \end{aligned}$$

and let G act on \(C_1\) by \(\sigma (z, w_2)=(z, \omega ^2 w_2)\). We consider the affine model

$$\begin{aligned} F_1=\{(t_3,x)\in {{\mathbb {C}}}^2| x^3=t_3(t_3+1)\} \end{aligned}$$

of the Fermat curve F as before. The rational map

$$\begin{aligned} \begin{aligned} C_1\times F_1&\rightarrow S_2, \\ (z, w_2, t_3, x)&\mapsto (z, t_3, \frac{x}{w_2}). \end{aligned} \end{aligned}$$

gives a G-birationally equivalence between \(C_1\times F_1 /G\) and \(S_2\). Moreover, we see the smooth projective model of \(C_1\) is isomorphic to C, the cyclic triple cover of \({{\mathbb {P}}}^1\) branched along the six points \(\{0,1,\infty , a, b, c\}\). By combining all of the birational equivalences above, we obtain S is G-birationally equivalent to \(C\times F/ G\). \(\square \)

Now it is direct to see that, by combining Lemma A.1 and Lemma A.2, we obtain the following birational equivalence.

Proposition A.3

Given distinct \(a,b,c\in {{\mathbb {C}}}\backslash \{0,1\}\), Rohde’s singular Calabi-Yau threefold \(X^{'}=W\times F /G\) is birational to X, which is the cyclic triple cover of \({{\mathbb {P}}}^3\) branched along \(\sum _{i=0}^5 H_i\) with \(H_i\) defined by

$$\begin{aligned} H_i: X_i=0 \ (0\le i\le 3), \ \ H_4: \sum _{i=0}^3X_i=0, \ \ H_5: X_0+a X_1+b X_2+c X_3=0. \end{aligned}$$

Note that to give distinct \(a,b,c\in {{\mathbb {C}}}\backslash \{0,1\}\) is equivalent to give six distinct points \(\{0, 1, \infty , a, b, c\}\) in \({{\mathbb {P}}}^1\). Since the moduli space \({\mathfrak {M}}_{1,6}\) of ordered six distinct points in \({{\mathbb {P}}}^1\) is isomorphic to the moduli space \({\mathfrak {M}}_{3, 6}\) of ordered six hyperplane arrangements in general in \({{\mathbb {P}}}^3\), the following birational equivalence is a direct consequence of Proposition A.3.

Proposition A.4

Rohde’s Calabi-Yau family \({\mathcal {X}}^{'}\rightarrow {\mathfrak {M}}_{1,6}\) is birationally equivalent to the universal family \({\mathcal {X}}_{AR}\rightarrow {\mathfrak {M}}_{3, 6}\) of cyclic triple covers of \({{\mathbb {P}}}^3\) branched along six hyperplanes in general position.

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Sheng, M., Xu, J. A Global Torelli Theorem for Certain Calabi-Yau Threefolds. Commun. Math. Stat. (2022). https://doi.org/10.1007/s40304-021-00283-2

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Keywords

  • Global Torelli theorem
  • Calabi-Yau threefolds
  • Hyperplane arrangements

Mathematics Subject Classification

  • 32G20
  • 14C34
  • 14J32