Products of three elliptic curves
Consider the abelian threefold
$$\begin{aligned} A{:}{=} E_1 \times E_2 \times E_3 , \quad E_\alpha = \mathbf {C}/\Lambda _\alpha , \, \text { with } \quad \Lambda _\alpha =\mathbf {Z}\oplus \mathbf {Z}\tau _\alpha ,\, \alpha =1,2,3. \end{aligned}$$
Using for a fixed elliptic curve \(E=\mathbf {C}\mathbf {Z}\oplus \tau \mathbf {Z}\) the involutions
$$\begin{aligned} \iota _E : z\mapsto -z, \quad t_E : z\mapsto - z+{\frac{1}{2}} ,\quad {\tau }_E : z \mapsto - z+ {\frac{1}{2}\tau }, \end{aligned}$$
we obtain three involutions on A
$$\begin{aligned} \begin{aligned} \iota _\alpha&=\iota _{E_\alpha } \\ \iota _{\alpha \beta }&= t_{E_\alpha } t_{E_\beta },\\ \iota _{123}&= \tau _{E_1}\tau _{E_2}\tau _{E_3}. \end{aligned} \end{aligned}$$
(1)
and we consider the group \((\mathbf {Z}/2\mathbf {Z})^6\) operating on A as
$$\begin{aligned} G_0 {:}{=} \langle \, \iota _1, \iota _2, \iota _3 ,\iota _{12} , \iota _{13}, \iota _{123} \, \rangle . \end{aligned}$$
Lemma 3.1
The action of \(G_0\) on holomorphic 1-forms of A is given by
Form
|
\(\iota _1\)
|
\(\iota _2\)
|
\(\iota _3\)
|
\(\iota _{12}\)
|
\(\iota _{13}\)
|
\(\iota _{23}\)
|
\(\iota _{123}\)
|
---|
\(dz_1\)
|
\(- \)
|
\(+\)
|
\(+\)
|
−
|
−
|
\(+\)
|
\(- \)
|
\(dz_2\)
|
\(+ \)
|
−
|
\(+\)
|
\(- \)
|
\(+\)
|
\( -\)
|
\(- \)
|
\(dz_3\)
|
\(+ \)
|
\(+\)
|
−
|
\(+ \)
|
−
|
\( -\)
|
\( -\)
|
Consider now the symmetric line bundle \({\mathscr {L}}_A^2\) where
$$\begin{aligned} {\mathscr {L}}_A:={\mathscr {O}}_{E_1}( {\mathscr {L}}_0) \boxtimes {\mathscr {O}}_{E_2}( {\mathscr {L}}_0) \boxtimes {\mathscr {O}}_{E_3}( {\mathscr {L}}_0), \end{aligned}$$
and set
$$\begin{aligned} H^0({\mathscr {L}}_A^2) = M_{E_1} \boxtimes M_{E_3} \boxtimes M_{E_3}. \end{aligned}$$
By Lemma 2.2 this is a representation space for \(G_0\) which admits a basis of simultaneous eigenvectors. If {\(\theta ^1_{E_j}, \theta ^2_{E_j}\}\), denotes the basis of Lemma 2.2, for \(M_{E_j}\), \(j=1,2,3\), their products give 8 basis vectors as follows.
$$\begin{aligned} \theta _{j_1j_2j_3}= \theta _{E_1}^{j_1} \cdot \theta _{E_2}^{j_2} \cdot \theta _{E_3}^{j_3},\quad j_k\in {1,2}. \end{aligned}$$
The next result is a consequence.
Lemma 3.2
The space \(H^0({\mathscr {L}}^2_A)\) is the \(G_0\)-representation space which on the basis \(\{ \theta _{j_1j_2j_3}\}\), \(j_1,j_2,j_3\in \{1,2\}\), is given as follows
Element
|
\(\iota _1\)
|
\(\iota _2\)
|
\(\iota _3\)
|
\(\iota _{12}\)
|
\(\iota _{13}\)
|
\(\iota _{23}\)
|
\(\iota _{123}\)
|
---|
\(\theta _{111}\)
|
\(+ \)
|
\(+\)
|
\(+\)
|
\(+ \)
|
\(+\)
|
\(+\)
|
−
|
\(\theta _{211}\)
|
\(+ \)
|
\(+\)
|
\(+\)
|
\(- \)
|
−
|
\(+\)
|
\( +\)
|
\(\theta _{121}\)
|
\(+ \)
|
\(+\)
|
\(+\)
|
−
|
\(+\)
|
−
|
\( +\)
|
\(\theta _{112}\)
|
\(+ \)
|
\(+\)
|
\(+\)
|
\(+ \)
|
−
|
−
|
\( +\)
|
\(\theta _{1\, 2\, 2}\)
|
\(+ \)
|
\(+\)
|
\(+\)
|
\(- \)
|
−
|
\(+\)
|
−
|
\(\theta _{2\, 1\, 2 }\)
|
\(+ \)
|
\(+\)
|
\(+\)
|
\(- \)
|
\(+\)
|
−
|
−
|
\(\theta _{221}\)
|
\(+ \)
|
\( +\)
|
\(+\)
|
\(+\)
|
−
|
−
|
−
|
\(\theta _{222}\)
|
\(+ \)
|
\(+\)
|
\(+\)
|
\(+\)
|
\(+\)
|
\(+\)
|
\( +\)
|
Hypersurfaces of abelian threefolds and involutions
Let A be an abelian variety of dimension three and \({\mathscr {L}}\) a principal polarization so that \({\mathscr {L}}^3=3! =6 \) and let \(i: X\hookrightarrow A\) be a smooth surface given by a section of of \({\mathscr {L}}^{\otimes 2}\). The Lefschetz’s hyperplane theorem gives:
Lemma 3.3
Suppose that \(\iota :A\rightarrow A\) is an involution which acts on \(H^0(\Omega ^1_A)\) with p eigenvalues 1 and \(n=3-p\) eigenvalues \(-1\). Suppose also that \(\iota \) preserves X and acts without fixed points on X. Then we have
$$\begin{aligned} \mathop {\mathrm{Tr}}\nolimits (\iota ) |H^2_{\mathrm{var}}(X) = -29 +8p(4-p) = {\left\{ \begin{array}{ll} -\,29 &{} \text {for } \quad p=0\\ -\,5 &{} \text {for }\quad p=1 \\ \,\,3 &{} \text {for }\quad p=2 \\ \,-5 &{} \text {for } \quad p=3. \end{array}\right. } \end{aligned}$$
Proof
The assumption implies that
$$\begin{aligned} \mathop {\mathrm{Tr}}\nolimits (\iota )|H^1(A) = 4p - 6= {\left\{ \begin{array}{ll} -\,6 &{} \text {for } \quad p=0\\ -\,2 &{} \text {for } \quad p=1 \\ \, 2 &{}\text {for }\quad p=2\\ \, 6&{} \text {for } \quad p=3, \end{array}\right. } \end{aligned}$$
and
$$\begin{aligned} \mathop {\mathrm{Tr}}\nolimits (\iota )| H^2(A)= 8p(p-3)+15= {\left\{ \begin{array}{ll} 15 &{} \text {for }\quad p=0\\ -\,1 &{} \text {for }\quad p=1 \\ {-\,1}&{} \text {for } \quad p=2 \\ \, 15&{} \text {for }\quad p=3. \end{array}\right. } \end{aligned}$$
If \(\iota \) preserves X and acts without fixed points on X, Lefschetz’ fixed point theorem gives \(0= 2 - 2 \mathop {\mathrm{Tr}}\nolimits (\iota ) | H^1(X) + \mathop {\mathrm{Tr}}\nolimits (\iota ) | H^2(X)= 2 - 2\mathop {\mathrm{Tr}}\nolimits (\iota ) | H^1(A) + \mathop {\mathrm{Tr}}\nolimits (\iota ) | H^2(A) + \mathop {\mathrm{Tr}}\nolimits (\iota ) |H^2_\mathrm{var}(X) ,\) and so the above calculation immediately gives the desired result. \(\square \)
In order to calculate the invariants on X, let me first consider the holomorphic two-forms in detail.
Lemma 3.4
-
1.
One has
$$\begin{aligned} h^{0,2}_{\mathrm{var}} (X) = 7,\quad h^{0,2}_{\mathrm{fix}} (X)= 3. \end{aligned}$$
-
2.
If \(X=\{\theta _0=0\}\), the variable holomorphic 2-forms are the Poincaré-residues along X of the meromorphic 3-forms on A given by expressions of the form
$$\begin{aligned} \frac{\theta }{\theta _0} dz_1\wedge dz_2\wedge dz_3 \end{aligned}$$
with \(\theta \) a theta-function on A corresponding to a section of \({\mathscr {L}}^{\otimes 2}\), and where \(z_1,z_2,z_3\) are holomorphic coordinates on \(\mathbf {C}^3\).
-
3.
Suppose \(\iota \) acts with the character \(\epsilon \in \{\pm 1\}\) on holomorphic three forms. Let (p, n) be the dimensions of the invariant, resp. anti-invariant sections of \({\mathscr {L}}^{\otimes 2}\). Then \(\dim H^{2,0}_{\mathrm{var,{\epsilon }}}(X)= p-1\) if \( \epsilon =1 \) and \(=p\) otherwise.
Proof
Consider the Poincaré residue sequence
In cohomology this gives
One sees that image of the residue map coincides with \(\ker {\left( H^0(\Omega ^2_X) \rightarrow H^1(\Omega ^3_A)\right) }\), which is the (2, 0)-part of the variable cohomology, by definition equal to
. Since \(H^0(\Omega ^3_A(X))= H^0({\mathscr {L}}^{\otimes 2})\) the assertion 1. follows.
-
2.
This is clear.
-
3.
This follows directly from (4). \(\square \)
Corollary 3.5
The invariants of X are as follows.
\(b_1\)
|
\(b_2^{\mathrm{var}}=(h^{2,0}_{\mathrm{var}},h^{1,1} _{\mathrm{var}},h^{0,2} _{\mathrm{var}} )\)
|
\( b_2^{\mathrm{fix}}=(h^{2,0}_{\mathrm{fix}},h^{1,1} _{\mathrm{fix}},h^{0,2} _{\mathrm{fix}} ) \)
|
---|
6
|
\(43=(7,29,7)\)
|
\(15=(3,9,3)\)
|
Proof
Equation (2) gives \(b_1(X)=b_1(A)=6\). To calculate \(b_2(X)\) we observe that \(c_1(X)= -2 {\mathscr {L}}|_X\) and \(c_2(X)= 4 {\mathscr {L}}^2|_X\) so that
$$\begin{aligned} c_1^2(X)=c_2(X)=4 {\mathscr {L}}^2|_X=8 {\mathscr {L}}^3 =48. \end{aligned}$$
Since \(c_2(X)= e(X)=2- 2b_1(X)+ b_2(X)=48\), it follows that \(b_2(X)= 58\). By (3) one has \(b_2^{\mathrm{fix}}(X)= b_2(A)=15\) and so \(b_2^{\mathrm{var}}(X)= 43\). Since \(h^{2,0}_\mathrm{var}=7\), the invariants for X follow. \(\square \)
Burniat hypersurfaces
A Burniat hypersurface of \(A=E_1\times E_2\times E_3\) is a surface which is invariant under a subgroup \(G\subset G_0\) generated by 3 commuting involutions and which acts freely on X. Each of the involutions is a product of the involutions (1). The quotient \(Y=X/G\) is called a generalized Burniat surface. In [1] one finds a list of 16 types of such surfaces, denoted \(\mathscr {S}_1,\dots ,\mathscr {S}_{16}\). All of the surfaces are of general type with \(c_1^2=6\), \(p_g=q\) and \(q=0,1,2,3\) and hence \(e=6= 2 -4q +b_2\) so that \(b_2=(p_g,h^{11},p_g)= (q, 4+2q, q)\). There are 4 families with \(q=0\) and one of these, \(\mathscr {S}_2\) gives the classical Burniat surfaces from [4]. See Table 1.
Table 1 Burniat hypersurfaces
The last column of this table gives the action of the three generators \((g_1,g_2,g_3)\) on \(H^0(\Omega ^3_A)\). It is calculated using Lemma 3.1. If an involution acts as the identity, there appears a “\(+\)” in the corresponding entry and else a “−”; e.g. \((+,-,-)\) means that \(g_1=\mathop {\mathrm{id}}\nolimits \) but \(g_2=g_3=-\mathop {\mathrm{id}}\nolimits \).
Table 2 Action on forms and invariants of the generalized Burniat surfaces
In Table 2 the character spaces for the action on the forms coming from A is given. It is calculated from the description of the generating involutions as given in Table 1 and the known action of 1-forms as given in Lemma 3.1. In Table 2 we use the shorthand \(\mathbf {1}\) for \((+++)\); the last two columns give the Hodge numbers \((h^{2,0},h^{1,1},h^{0,2} )\). From the first column of this table one finds the trace of the action of these generators on \(H^0(\Omega ^1_A)\), or, alternatively, the dimensions of the eigenspaces for the eigenvalues \(+1\) and \(-1\). Writing the dimensions of the \((+)\)-eigenspaces as a vector according to the group elements written in the order \((1, g_1,g_2,g_3,g_1g_2,g_1g_3,g_2g_3,g_1g_2g_3)\) yields the type \((3,t_1,t_2,t_3,\dots )\in \mathbf {Z}^8\) of the group action. This gives the first row in Table 3 below. Using Lemma 3.3, this table enables to find the multiplicity of \(\chi _A\) in \(H^2_{\mathrm{var}}(X)\).Footnote 4
Lemma 3.6
For each of the families \(\mathscr {S}_3\)–\(\mathscr {S}_{16}\) the character of \(\chi _A\) appears with non-zero multiplicity in the variable cohomology.Footnote 5
Proof
For each of the families \(\mathscr {S}_3,\mathscr {S}_4, \mathscr {S}_{11}, \mathscr {S}_{12}\) and \(\mathscr {S}_{16}\) one has \(H^0(A,\Omega ^3_A)=(+++)\) and \({\dim }H^{1,1}_\mathrm{var}(Y)= \dim H^{1,1}_{\mathrm{var, +++}}(X) =1\), as one sees from Table 2.
For the other families we argue as follows. In each case, \(g\in G\), \(g\not =1 \) act freely on X and so one can apply Lemma 3.3 to find \(\mathop {\mathrm{Tr}}\nolimits g|H^2_{\mathrm{var}}(X)\), given the dimension p(g) of the \((+1)\)-eigenspace of \(H^0(\Omega ^1_A)\). This type is given in Table 3. The next column gives the corresponding trace vector. Then follows the trace vector of \(\chi _A\). Now apply the trace formula for the multiplicity of an irreducible representation inside a given representation (see e.g. [12, §2.3]): just take the “dot” product of the two trace vectors and divide by the order of the group. Let me do this explicitly for the family \(\mathscr {S}_5\). The trace vector for \(H^2_{\mathrm{var}}(X)\) is \((43,-5,-5,{3},-5,3,3,3)\), the first number being \(\dim H^2_{\mathrm{var}}(X)\). The representation \(\chi _A=(++-)\) has trace vector \((1,1,1,-1,1,-1,-1,-1)\) and the trace formula reads
$$\begin{aligned} \frac{1}{8} (43-5-5{-3}-5-3-3-3)= {2}. \end{aligned}$$
\(\square \)