A motivic study of generalized Burniat surfaces

Generalized Burniat surfaces are surfaces of general type with $p_g=q$ and Euler number $e=6$ obtained by a variant of Inoue's construction method for the classical Burniat surfaces. I prove a variant of the Bloch conjecture for these surfaces. The method applies also to the so-called Sicilian surfaces introduced by Bauer, Catanese and Frapporti. This implies that the Chow motives of all of these surfaces are finite-dimensional in the sense of Kimura.

= 0. These generalized Burniat surfaces Y = X /G are all quotients of X by a freely acting abelian group G (Z/2Z) 3 and where (X , G) is a so called Burniat hypersurface pair (X , G): X is a hypersurface in a product A of three elliptic curves having at most nodes 123 as singularities and G is an abelian group acting freely on A and leaving X invariant. The surface X is also called Burniat hypersurface. These come in 16 families, enumerated S 1 -S 16 . The classical Burniat surface belongs to the 4-parameter family S 2 . Also the family S 1 is 4dimensional. The remaining families have only 3 parameters coming from varying the elliptic curve. This implies that the equation of X in these cases is uniquely determined, contrary to the first two where there is a pencil of hypersurfaces invariant under G. The surfaces Y have at most nodal singularities. For simplicity I assume in this note that Y , and hence X is smooth, which is generically the case. However, none of the arguments is influenced by the presence of nodal singularities.
In [9] it has been remarked that the main theorem of loc. cit. yields the Bloch conjecture for the classical Burniat surfaces. The goal of this paper is to apply the same methods to all Burniat hypersurfaces. In particular, one obtains a short proof of the Bloch conjecture in the appropriate cases.
To state the result, let me recall that the Chow motive 1 h(X ) is the pair (X , ) where ⊂ X × X is the diagonal considered as a (degree 0) self-correspondence of X . As a selfcorrespondence it is an idempotent in the ring Corr 0 (X ) of degree 0 self-correspondences. If a finite group G acts on X , any character χ of the group defines an idempotent where g is the graph of the action of g on X . The pair (X , π χ ) is the motive canonically associated to the character χ. Note that the trivial character gives the motive h(X /G) of the Burniat hypersurface. The main result now reads as follows: For the families S 11 , S 12 and S 16 this means that ker(i * : CH 0 (X ) → CH 0 (A)) = 0. 3 As shown in [1] the families S 11 and S 12 give two divisors in a 4-dimensional component of the Gieseker moduli space.
The above method applies also to the surfaces in this component, the so-called Sicilian surfaces so that the result for S 11 and S 12 is valid for these as well. See Remark 4.3.

Motives
A degree k (Chow) correspondences from a smooth projective variety X to a smooth projective variety Y is a cycle class A correspondence of degree k induces a morphism on Chow groups of the same degree and on cohomology groups (of double the degree). Correspondences can be composed and these give the morphisms in the category of Chow motives. Let me elaborate briefly on this but refer to [10] for more details.
Precisely, an effective Chow motive consists of a pair (X , p) with X a smooth projective variety and p a degree zero correspondence which is a projector, i.e., p 2 = p. Morphism between motives are induced by degree zero correspondences compatible with projectors. This procedure defines the category of effective Chow motives. Every smooth projective variety X defines a motive and a morphism f : X → Y between smooth projective varieties defines a morphism h(Y ) → h(X ) given by the transpose of the graph of f . One can also use correspondences of arbitrary degrees provided one uses triples (X , p, k) where p is again a projector, but a morphism f : (X , p, k) → (Y , q, ) is a correspondence of degree − k compatible with projectors. Such triples define the category of Chow motives. It should be recalled that motives, like varieties have their Chow groups and cohomology groups: Kimura [8] has introduced the concept finite-dimensionality for motives. If the motive of a surface S is finite-dimensional, then the Bloch conjecture holds for any submotive M of S with h 2,0 (M) = 0. This is the motivation for considering the variable cohomology. In [11] it is shown that there is indeed a submotive of S whose cohomology is the variable motive.

A criterion for finite dimensional motives
The general situation of [9] concerns smooth d-dimensional complete intersections X inside a smooth projective manifold M of dimension d + r for which Lefschetz' conjecture B(M) holds. This conjecture is known to hold for projective space and for abelian varieties and so in particular for the situation in this note.
Recall also (see e.g. [5,Ch. 3.2]) that in this situation, with i : X → M the inclusion, the fixed and variable cohomology is defined as follows.
and that one has a direct sum decomposition which is orthogonal with respect to the intersection product. In [11] I explained that validity of B(M) implies the existence of a motive (X , π var ) such that π var induces projection onto variable cohomology. The main input is the special case of [9, Thm. 6.5 and Cor. 6.6] for surfaces inside a threefold. It reads as follows.

Elliptic curves
Let me recall the relevant facts about theta functions on an elliptic curve E with period lattice generated by 1 and τ ∈ h. Points in the elliptic curve referred to by the standard coordinate z ∈ C and the corresponding line bundle by L z . It is the bundle with H 0 (E, L z ) = Cϑ z , ϑ z a theta-function with simple zeros in the points z + only. Let t u : z → z + u be a translation of E. Then L z t * z L 0 . If one takes for z one of the four two-torsion points ∈ {0, 1 2 , 1 2 τ, 1 2 + 1 2 τ } of E, the corresponding line bundles L have the four classical theta functions ϑ 1 , ϑ 2 , ϑ 3 , ϑ 4 respectively as sections. See e.g. [6, Appendix A, Table 16 ] for the definitions. Set

Lemma 2.2 (i) The bundle L 2 0 is a symmetric line bundle and all its sections are symmetric. (ii) The translations t define a faithful action of
0 for all two-torsion points , the functions ϑ 2 j define sections of the same bundle L 2 0 . The sections ϑ 2 j , j = 1, 2, 3, 4 are characterized by having a double zero at exactly one of the four 2-torsion points. This shows in particular that the action of the group {t , a 2-torsion point} is faithful on M E . (iii) It follows that there is a basis of two sections of L 2 0 consisting of simultaneous eigenvectors for this action. Since the action is faithful, the character decomposition must be (+−), (−+).

Consider the abelian threefold
Using for a fixed elliptic curve E = CZ ⊕ τ Z the involutions we obtain three involutions on A and we consider the group (Z/2Z) 6 operating on A as

Lemma 3.1 The action of G 0 on holomorphic 1-forms of A is given by
By Lemma 2.2 this is a representation space for G 0 which admits a basis of simultaneous eigenvectors. If {θ 1 E j , θ 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.
The next result is a consequence.

Hypersurfaces of abelian threefolds and involutions
Let A be an abelian variety of dimension three and L a principal polarization so that L 3 = 3! = 6 and let i : X → A be a smooth surface given by a section of of L ⊗2 . The Lefschetz's hyperplane theorem gives: If ι preserves X and acts without fixed points on X , Lefschetz' fixed point theorem gives 0 = 2−2 Tr(ι)|H 1 (X )+Tr(ι)|H 2 (X ) = 2−2 Tr(ι)|H 1 (A)+Tr(ι)|H 2 (A)+Tr(ι)|H 2 var (X ), and so the above calculation immediately gives the desired result.

Lemma 3.3 Suppose that ι : A → A is an involution which acts on H
In order to calculate the invariants on X , let me first consider the holomorphic two-forms in detail. Proof Consider the Poincaré residue sequence One sees that image of the residue map coincides with ker H 0 ( 2 , which is the (2, 0)-part of the variable cohomology, by definition equal to ker H 2 (X ) Proof Equation (2) gives b 1 (X ) = b 1 (A) = 6. To calculate b 2 (X ) we observe that c 1 (X ) = −2L| X and c 2 (X ) = 4L 2 | X so that Since h 2,0 var = 7, the invariants for X follow.

Burniat hypersurfaces
A Burniat hypersurface of A = E 1 × E 2 × E 3 is a surface which is invariant under a subgroup G ⊂ 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 S 1 , . . . , S 16 .
The last column of this table gives the action of the three generators (g 1 , g 2 , g 3 ) on H 0 ( 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 = id but g 2 = g 3 = − id.
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 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 ( 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 1 g 2 , g 1 g 3 , g 2 g 3 , g 1 g 2 g 3 ) yields the type (3, t 1 , t 2 , t 3 , . . . ) ∈ 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 χ A in H 2 var (X ). 4 Lemma 3.6 For each of the families S 3 -S 16 the character of χ A appears with non-zero multiplicity in the variable cohomology. 5 Proof For each of the families S 3 , S 4 , S 11 , S 12 and S 16 one has H 0 (A, 3 A ) = (+ + +) and dim H 1,1 var (Y ) = dim H 1,1 var,+++ (X ) = 1, as one sees from Table 2. For the other families we argue as follows. In each case, g ∈ G, g = 1 act freely on X and so one can apply Lemma 3.3 to find Tr g|H 2 var (X ), given the dimension p(g) of Table 2 Action on forms and invariants of the generalized Burniat surfaces

The main result
In this section I shall show that the main theorem 4.2 below follows upon application of Theorem 2.1. First an auxiliary result.  ---)). Proof This follows from the G-action on the basis θ j 1 , j 2 j 3 for H 0 (L 2 A ) which can be deduced from Lemma 3.2. I shall work this out for two cases: the family S 2 , and for the family S 6 . For S 2 we have g 1 = ι 1 ι 3 ι 23 , g 2 = ι 3 ι 13 and g 3 = ι 2 ι 23 and for S 6 we have g 1 = ι 2 ι 3 ι 123 , g 2 = ι 2 ι 3 ι 13 and g 3 = ι 3 ι 23 , and the action of these involutions is given in the following table.
Element var (X ) except maybe this missing character χ A . But its multiplicity has been calculated in Table 3. It is non-zero and so condition (3) holds as well.
(3) This is one of the assertions of Theorem 2.1.

Remark 4.3
Recall the following definition from [1]: a Sicilian surface is a minimal surface S of general type with numerical invariants p g (S) = q(S) = 1, c 2 1 (S) = 6 for which, in addition, there exists an unramified double coverŜ → S with q(Ŝ) = 3, and such that the Albanese morphismα :Ŝ → Alb(Ŝ) is birational to its image Z , a divisor in its Albanese variety with Z 3 = 12. In loc. cit. one finds the following explicit construction. Let T = C 2 / 2 , 2 = Z 2 ⊕ τ 1 Z ⊕ Zτ 2 be an Abelian surface with a (1, 2)-polarization L 2 and let E = C/ , = Z ⊕ τ Z be an elliptic curve. Consider the sections of the line bundle L = L ⊗2 0 L 2 on A:=E × T that are invariant under the action of the bi-cyclic group K generated by (e, a) → (e + 1 2 τ, −a + 1 2 τ 1 ) and (e, a) → (e + 1 2 , a + 1 2 τ 2 ). These sections define hypersurfaces X ⊂ A and the quotient Y = X /K is a Sicilian surface and all such surfaces are obtained in this way.