On the cohomology of surfaces with $p_g = q = 2$ and maximal Albanese dimension

In this paper we study the cohomology of smooth projective complex surfaces $S$ of general type with invariants $p_g = q = 2$ and surjective Albanese morphism. We show that on a Hodge-theoretic level, the cohomology is described by the cohomology of the Albanese variety and a K3 surface $X$ that we call the K3 partner of $S$. Furthermore, we show that in suitable cases we can geometrically construct the K3 partner $X$ and an algebraic correspondence in $S \times X$ that relates the cohomology of $S$ and $X$. Finally, we prove the Tate and Mumford-Tate conjectures for those surfaces $S$ that lie in connected components of the Gieseker moduli space that contain a product-quotient surface.


Introduction
. -Let S be a smooth projective complex surface with invariants p g (S) = q(S) = 2, and assume that the Albanese morphism α : S → A is surjective. The results of this paper are inspired by the following two observations: . The induced map on cohomology α * : H * (A, Z) → H * (S, Z) is injective. The orthogonal complement H 2 new = H * (A, Z) ⊥ ⊂ H * (S, Z) is a Hodge structure of weight 2 with Hodge numbers (1, n, 1), where n = h 1,1 (S) − 4. Such a Hodge structure is said to be of K type.
. Let S be a smooth projective complex surface with invariant p g (S ) = 1. Then Morrison [Mor ] showed that there exists a K surface X together with an isomorphism ι : H 2 (S , Q) tra → H 2 (X , Q) tra that preserves the Hodge structure, the integral structure, and the intersection pairing. (Here (_) tra denotes the transcendental part of a Hodge structure, that is, the orthogonal complement of the Hodge classes.) These observations lead to the following questions.
. Q u e s t i o n A . -Let S be as before.
Does there exist a K surface X together with an isomorphism ι : (H 2 new ) tra Q → H 2 (X, Q) tra that preserves (H) the Hodge structure, or (Z) the integral structure, or (P) the intersection pairing? arXiv:1901.00193v1 [math.AG] 1 Jan 2019 We give an affirmative answer to this question in theorem . , showing that there exists an X and ι that satisfy (H), (Z), and (P). The strategy of the proof is the same as for Morrison's result mentioned above.
. Q u e s t i o n B . -The Hodge conjecture predicts that if ι satisfies (H), then it is algebraic.
Do there exist X and ι satisfying (H) as above, such that ι is algebraic?
In general we are not able to answer this question. However, an intesting class of examples of the surfaces that we consider is formed by so-called product-quotients: these are surfaces birational to a surface (C × D)/G, where C and D are curves equipped with an action by a finite group G.
In these cases we give an affirmative answer to question B in theorem . . The strategy boils down to finding appropriate Prym varieties in the Jacobians of C and D, and taking for X the associated Kummer variety. . Q u e s t i o n C . -Since we are not able to settle question B in general, we may aim for something weaker, sitting in between question A.(H) and question B. We use the notion of motivated cycles introduced by André [And ] (see § . for details).

Do there exist X and ι satisfying (H) as above, such that ι is motivated?
Once again, we are not able to give an answer to this question in general. However, we give sufficient conditions for a positive answer to this question. For example, we show that to decide question C one may replace S with any other surface in the same connected component of the moduli space of surfaces of general type (theorem . ). In particular, question C has a positive answer for every surface S that lies in the same connected component as a product-quotient surface.
If question C has a positive answer for the surface S, then we also deduce that the Tate and Mumford-Tate conjectures hold for models of S over finitely generated subfields of C (corollary . ).
We summarise these results in the following theorem (the conjunction of theorems . , . and . and corollary . ).
. T h e o r e m . -Let S be a smooth projective complex surface with invariants p g (S) = q(S) = 2, and assume that the Albanese morphism α : S → A is surjective.
. Then there exists a K surface X and an isomorphism of Hodge structures ι : (H 2 new (S, Q)) tra → H 2 (X, Q) tra .
. If S is a product-quotient surface ( cf. definition . ) with group G, then there exist X and ι as above, and an algebraic cycle on S × X that induces ι.
/ . If S is in the same connected component of the Gieseker moduli space as a product-quotient surface, then ι is motivated (in the sense of André)  . S t ru c t u r e o f t h i s t e x t . -In the next section, "On the classification of surfaces with p g = q = 2", we very briefly recall what is known for the surfaces under consideration. It is important to stress that the classification of these surfaces is not yet complete. Hence we present the state of the art up to now. We shall pay particular attention to those surfaces which are product-quotients (here a classification theorem is available) recalling definitions, important properties and its associated group theoretical data. Furthermore, we recall what is known about their moduli space.
In section we discuss the existence of Hodge-theoretical K partners for all the surfaces with p g = q = 2, following Morrison's theory. In theorem . we prove point of theorem . . In section we discuss the problem to find a geometric description where possible of the The second author was partially supported by MIUR PRIN "Geometry of Algebraic Varieties" and also by GNSAGA of INdAM.
On the classification of surfaces with p g = q = 2 . -The classification of smooth projective complex surfaces with invariants p g = q = 2 is not complete, although there has been much progress in recent years. We give an overview of the current state of the art. Let S be a minimal surface of general type with p g = q = 2, so that χ(S) = 1 − q + p g = 1. Recall the following classical general inequalities: . These yield 4 ≤ K 2 S ≤ 9 under our assumptions. Except for the case K 2 S = 9, examples of such surfaces have been constructred for every value of K 2 S in this range. Since we are dealing with irregular surfaces, i.e., with q > 0, a useful tool to study them is the Albanese map. The Albanese variety of S is the q-dimensional variety Alb(S) := H 0 (S, Ω 1 S ) ∨ /H 1 (S, Z). By Hodge theory, Alb(S) is an abelian variety. For a fixed base point x 0 ∈ S, we define the Albanese morphism as The dimension of α(S) (≤ q(S) = 2) is called the Albanese dimension of S and it is denoted by Albdim(S). If Albdim(S) = 2, we say that S has maximal Albanese dimension. For surfaces with q(S) = 2, we have two possibilities: . Albdim(S) = 1 and α(S) is a smooth curve of genus 2; or . Albdim(S) = 2 and α is a generically finite cover of an abelian surface.
The first case is completely understood: we have a classification theorem, see [Pen ]. . -In the rest of this section we will describe the examples № = 3, 4, 6, 8, 9 in more detail.

/
The examples № = 1, 2, 5, 7 will not be treated in this paper for the following reasons. The surfaces in № = 1 do not have a surjective Albanese map, therefore this case falls outside the scope of this paper. For the surfaces in № = 5, at the time of writing, nothing is known about their moduli spaces. Despite the abundance of information about their moduli, for the surfaces in № = 2 and 7 none of the methods we develop here seem to work; principally this is due to the small dimensions of their moduli.
. D e f i n i t i o n . -Let G be a finite group acting on two compact Riemann surfaces C 1 , C 2 of respective genera g 1 , g 2 ≥ 2. Consider the diagonal action of G on C 1 × C 2 . In this situation we say for short: the action of G on C 1 × C 2 is unmixed. By [Cat ] we may assume w.l.o.g. that G acts faithfully on both factors. The minimal resolution S of the singularities of T = (C 1 × C 2 )/G, is called a product-quotient surface. If the action of G on the product C 1 × C 2 is free we will speak of surfaces isogenous to a product of unmixed type. In this case T is already smooth.
. D e f i n i t i o n . -Let C be a smooth projective curve, and let G be a finite subgroup of Aut(C) × Z/2. Assume that there are elements in G exchanging the two factors of C × C. In this case we say that the action is mixed.
The minimal resolution of singularities S −→ T = (C × C)/G is called a mixed surface. A surface isogenous to a product of mixed type is a mixed surface where G acts freely on C × C.
We denote by G 0 G the index two subgroup, i.e., the subgroup of elements that do not exchange the factors. In general the singularities of T are rather complicated. Assuming that G 0 acts freely, i.e., (C × C)/G 0 is a surface isogenous to a product, then T is smooth and we call it a semi-isogenous mixed surface.
. -The families of surfaces № = 1, 3 in table are all the surfaces isogenous to a product of umixed type (see [Pen ]) and a brief description of those with maximal Albanese dimension (hence №= ) can be read from table below. There one can find the genus of C 1 and C 2 , the group G, and the number branch points with multiplicity of the covering C i −→ C i /G for i = 1, 2.
Notice first that the curves C i /G are elliptic curves for i = 1, 2; second that for a complete description one needs a system of generators for the group G. The families of surfaces № = 6, 8 and 9 contain subloci of product-quotient surfaces. To briefly describe these particular members we add information about the singularities of (C 1 × C 2 )/G to table .
Table describes two families of mixed surfaces that are of interest to us. The surfaces described by the first row are surfaces isogenous to a product of mixed type. The semi-isogenous mixed surfaces T = (C × C)/G with p g (T ) = q(T ) = 2 and K 2 T > 0, form 9 families [CF ]. We will not be able to say anything about the existence of algebraic K partners for these surfaces (section ), but we will study them in section . Since the questions studied in section will turn out to be invariant under deformation we only need to study one subfamily per component of the Gieseker moduli space. Of the families mentioned above, are subfamilies of components of (2), (2) 4 × 1 2 (1, 1) 4 2 9 4 2 2 C 2 (2 2 ), (2 2 ) 4 × 1 2 (1, 1) 4 4 9 Table : Classification of product-quotient surfaces of unmixed type with p g = q = 2. In the column labeled η we give the number of irreducible components of the exceptional divisor of the minimal resolution of (C 1 × C 2 )/G. The last column (№) mentions the component of the Gieseker moduli space described in table that contains the family of product-quotient surfaces as a subfamily.
the Gieseker moduli space that also contain families of product-quotient surfaces that we already described (see [CF ] and [Pig ]). The only families for which this is not the case are listed in table ; they are families in № = 3, 4 described in table . Table : The two families of mixed surfaces with p g = q = 2 that are not subfamilies of components of the Gieseker moduli space that also contain families of product-quotient surfaces that we described before.
. -As already remarked, the surfaces in component № = 3 are all isogenous to a product. Hence the weak rigidity theorem of Catanese [Cat ] tells us that for each family their moduli space consists of one connected irreducible component in the subspace M 8,2,2 (M K 2 ,pg,q ) of the Gieseker moduli space of surfaces of general type M 8,1 (M K 2 ,χ ). Moreover each member of the family is isogenous to a product.
The property of preserving an isotrivial fibration is no longer true for the families № = 4, 6, 8, 9.
Indeed, their moduli space is bigger in some sense. To be precise let us first analyse the families of surfaces in table with K 2 S < 8. These families form an irreducible sublocus of M K 2 ,2,2 but they sit inside a bigger connected component. The first entry of table is again isogenous to a product, of mixed type. Hence its moduli space is 3-dimensional and is described by the weak rigidity theorem of Catanese [Cat ].
The moduli space of the surfaces relative to the second entry of table is described by Pignatelli and Polizzi in [PP ]. In this case the moduli space is a generically smooth, irreducible, open and normal subset of the Gieseker moduli space M 7,2,2 . For the general member of the family the Albanese surface is simple, but some specific surfaces admit an irrational fibration over an elliptic curve.

Hodge-theoretic K partners
. -Let S be a smooth projective complex surface with invariants p g (S) = q(S) = 2, and assume that the Albanese morphism α : S → A is surjective. Recall question A from the introduction:

preserves (H) the Hodge structure, or (Z) the integral structure, or (P) the intersection pairing?
In theorem . we give an affirmative answer to this question.
. -A Hodge lattice V is a free Z-module of finite rank, endowed with a polarised Hodge structure such that the polarisation on V makes V into a lattice-a symmetric bilinear form on a free Z-module of finite rank. In particular, the weight (as Hodge structure) of V is always even.
. N o tat i o n . -There is some risk of confusing notation: if V is a Hodge lattice, then V (k) may denote either the k-th Tate twist of the Hodge structure on V or it may denote the k-th twist of the lattice structure on V . In this paper we use the notation V (k) only for Tate twists of the Hodge structure.
. D e f i n i t i o n . -Let V be a Hodge lattice of K type (that is, the Hodge structure on V / . -Let Λ denote the even unimodular lattice E 8 −1 2 ⊕ U 3 . The lattice Λ goes by the name K lattice, since there is an isometry H 2 (X, Z) ∼ = Λ for every complex K surface X. For an . t + < s + and t − < s − ; and Proof . This is a slightly weaker form of theorem . . of [Nik ].
The following corollary is part of an observation by Morrison, see corollary . of [Mor ].
. C o ro l l a ry . -Let M be an even lattice with signature (2, n) for some integer 0 ≤ n ≤ 8.
Then there exists a unique primitive embedding M → Λ into the K lattice introduced above.
Define → Ω(Λ) the Hodge structure on Vb maps to a point x ∈ Ω(Λ). By the surjectivity of the period map for K surfaces there is exists a complex K surface X, with H 2 (X, Z) ∼ = Λ, such that the Hodge structure on H 2 (X, Z) corresponds to x.
Let d ∈ Z and K ⊂ SO(Λ d )(Ẑ) be such that X admits a primitive polarisation of degree 2d with full level K-structure. This means that [X] ∈ F full 2d,K,C . Since L has signature (2, n) we get a primitive embedding L → Λ d . This yields a diagram , where the latter is the Shimura variety parameterising polarised Hodge structures on Λ d ⊗ Q with a level K-structure.
We now have a diagram where the bottom right rectangle is cartesian. By our choice of d and K, we know that [X] ∈ F full 2d,K,C and b ∈ B are in the image of B • under the respective maps. In particular B • is non-empty.
Pulling back the universal family of K surfaces from F full 2d,K,C to B • we end up with a K space f : X → B • and a morphism of variations of Hodge structures V| B • → R 2 f * Z. By construction it is fibrewise a primitive embedding and a Hodge isometry on the transcendental lattices. .
L e m m a . -Let S be a smooth projective complex surface with invariants p g = q = 2. Let α : S → A be the Albanese morphism, and assume that α is surjective.
Finally, b 2 is the sum of the ranks of H 2 new and H 2 (A, Z). The latter has rank 6, and we conclude that .
tra is by definition perpendicular to all Hodge classes. The result follows from corollary . and proposition . .

Algebraic K partners
Let S be a smooth projective complex surface with invariants p g (S) = q(S) = 2, and assume that the Albanese morphism α : S → A is surjective. In this section we attempt to answer question B of the introduction: Does there exist a K surface X together with an isomorphism ι : (H 2 new ) tra Q → H 2 (X, Q) tra that is algebraic?
Since theorem . provides an affirmative answer to question A, the Hodge conjecture predicts a positive answer to question B as well. In theorem . we show that this is indeed the case for certain surfaces that are product-quotients (cf. definition . ). If the essence of the proof must be captured in one sentence, it would be the following: the algebraic correspondence inducing ι is built from the Kummer K surface associated with a suitable 2-dimensional isogeny factor of the product of the Jacobians of the curves that are used in the construction of the product-quotient surface S. of [Sch ]). There exists a functor h : SmPr op /C → M rat from the category of smooth projective varieties over C to the category of Chow motives. If X is a smooth projective variety, and G is a finite group that acts on X, then we define h(X/G) = h(X) G . This leads to a satisfactory theory of motives of quotient varieties, as is explained in [BN ].
We denote with CH i (X) (resp. CH i (M )) the i-th Chow group of a smooth projective variety X (resp. a motive M ∈ M rat ). In general, it is not known whether the Künneth projectors π i are algebraic, so it does not (yet) make sense to speak of the summand h i (X) ⊂ h(X) for an arbitrary smooth projective variety X/C. However, a so-called Chow-Künneth decomposition does exist and where η is the number of exceptional divisors introduced in the minimal desingularization of the quotient surface.
/ Proof . Let η denote the number of exceptional divisors introduced in the minimal desingularization of the quotient surface. Recall that for the surfaces that we are interested in we gave the value of η in table . Observe that h 2 (S) = h 2 ((C 1 × C 2 )/G) ⊕ E. Notice that if the action of G on C 1 × C 2 is free then E = 0. By the Künneth formula we obtain By definition we have h 2 ((C 1 × C 2 )/G) = h 2 (C 1 × C 2 ) G ; and since G acts trivially on h 0 (C i ) and h 2 (C i ), we get the result.
. R e m a r k . -Note that we can further decompose the Chow motive Z of proposition .
whereĜ is the set of isomorphism classes of irreducible representations of G over Q, χ 1 is the trivial representation, and (_) (W ) denotes the W -isotypical component.
. If G is a finite group, and W is equipped with a finite-dimensional representation of G, then we may view the representation G → Aut(W ) as an action of G on the Chow motive W .
. -We now state the main theorem of this section, which gives a partial answer to question B of the introduction. The rest of this section is dedicated to its proof. We refer to § . for an explicit computation that illustrates the general argument of this proof.
. T h e o r e m . -Let S be a minimal surface of general type with p g = q = 2, of maximal Albanese dimension isogenous to a product of unmixed type. Then there exists a K surface X and a correspondence in S × X that induces an isomorphism between H 2 (X, Q) tra and H 2 (S, Q) tra new .
. -The proof is done in several steps, and will be completed in § . . Following proposition . and remark . we decompose h 2 (S) = U ⊕ Z 1 ⊕ Z 2 ⊕ E. First of all we see that for i = 1, 2 the quotients C i /G =: E i are elliptic curves. The product E 1 × E 2 is the Albanese variety of S, and we have h 2 (E 1 × E 2 ) = U ⊕ Z 1 . Secondly, let η be the number of irreducible components of the exceptional divisor of the minimal resolution of the quotient surface (see tables and ; if there is no branch locus, then η = 0). This gives E = Q(−1) η . For the purpose of this theorem we are interested in the remaining term Z 2 . To find an algebraic K partner X for S, we will find an abelian surface A as isogeny factor of J(C 1 ) × J(C 2 ), such that Z 2 = h 2 (A) tra ⊕ Q(−1) k for some k ∈ Z ≥0 . We may then take the minimal resolution of singularities of the Kummer / surface Km(A) = A/ −1 for the K surface X. To find the isogeny factor A, we proceed by decomposing (up to isogeny) the Jacobians J(C i ) of C i for i = 1, 2 as products of simple abelian varieties following [PR ].
. -Now let A be an abelian variety of dimension g with a faithful action of a finite group G. There is an induced homomorphism of Q-algebras

Any element α ∈ Q[G] defines an abelian subvariety
where m is some positive integer such that mρ(α) ∈ End(A). This definition does not depend on the chosen integer m.
We will now describe the so-called isotypical decomposition of the abelian variety A with group action by G. Begin with the decomposition of Q[G] as a product of simple Q-algebras Q 1 × · · · × Q r . The factors Q i correspond canonically to the rational irreducible representations W i of the group G, because each one is generated by a unit element e i ∈ Q i which may be considered as a central idempotent of Q[G].
The corresponding decomposition of 1 ∈ Q[G], 1 = e 1 + · · · + e r induces an isogeny, via ρ above, which is given by addition. Note that the components A ei are G-stable complex subtori of A with Hom G (A ei , A ej ) = 0 for i = j. The decomposition ( . . ) is the isotypical decomposition mentioned above.
. -The isotypical components A ei can be decomposed further, using the decomposition of Q i into a product of minimal left ideals. Fix an i ∈ {1, . . . , r}, and let W i be the irreducible rational representation of G corresponding to the idempotent e i . We will now recall some facts from representation theory; see § . of [Ser ] for details.
Write D i for the simple algebra End G (W i ), and observe that Q i = Mat ni (D • i ) is a matrix algebra of degree n i over the opposite algebra, for some n i ∈ Z ≥1 . Recall that the Schur index of W i is the degree m i of D i over its centre. If χ i is the character of one of the irreducible summands such that e i = π i,1 + · · · + π i,ni .
(We warn the reader that the π i,j are not G-equivariant, and hence the abelian subvarieties A πi,j are not G-stable.) The abelian subvarieties A πi,j are mutually isogenous for j = 1, . . . , n i . Let . -While the factors in ( . . ) are not necessarily easy to determine, we may compute their dimension in the case of a Jacobian variety. Let C be a compact Riemann surface equipped with an action of a finite group G and consider the induced action of G on J(C). Define V to be the representation of G on H 1 (X, Z) ⊗ Z Q. We use the same notation as at the beginning of this section, so the quotient C/G has genus g 0 and the cover π : C → C/G has r branch points {q 1 , . . . , q r } where each q i has corresponding monodromy g i . The tuple (g 1 , . . . , g r ) is called the generating vector for the action [Bro ].
We now copy equation . from [Bro ], and explain the notation afterwards: the Hurwitz character χ V associated to V is Here χ 1 is the trivial character on G, and ρ 1 * is the character of the regular representation. The With this definition of χ V in place, we have the following equality where ψ i denotes the character of the Q-irreducible representation of G corresponding to W i . See . -We will now complete the proof of theorem . . We may calculate the dimension of the B i 's for each class of surfaces in the statement, either by hand or using the MAGMA script as in [PR ]. The result of this computation is given in table .
It is a coincidence that in the table all the characters that appear are actually self-dual and defined over Q. One can check that for each row in the table, there is only one character χ that appears in column char(C 1 ) such that the dual character χ ∨ appears in column char(C 2 ).  Let W be the irreducible rational representation of G that corresponds to χ. We will complete the proof by a case distinction. First assume that G = Q 8 . In this case, one may check that the Schur index of W is 1, and in fact D = End G (W ) = Q. In other words, we are in the situtation of remark . . Let L 1 be a reduced factor of J(C 1 ) corresponding with χ, and denote with L 2 a reduced factor of J(C 2 ) that corresponds to χ ∨ . Thus we have as motives with an action of G. Consequently, we find We conclude that Z 2 ∼ = h 1 (L 1 ) ⊗ h 1 (L 2 ) ∼ = h 2 (L 1 × L 2 ) tra ⊕ Q(−1) ⊕k for some k ∈ Z ≥0 , and thus L 1 × L 2 is the 2-dimensional isogeny factor A of J(C 1 ) × J(C 2 ) that we are looking for. .
T h e c a s e G = Q 8 . -In the case where G = Q 8 , we find that D = End G (W ) = H which has Schur index 2. In this case we cannot use the methods employed so far to prove that the Jacobian is isogenous to a product of elliptic curves as in all the other cases.
Nevertheless, in [FPP ] it is proven that in this case the curves C 1 and C 2 , which are of genus g = 3, admit a bigger automorphism group. Indeed, their automorphism group is isomorphic to (C 4 × C 2 ) C 2 , which readily contains Q 8 . More precisely, in [FPP ] it is shown that the curves C of genus g = 3 and with automorphism Q 8 and (C 4 × C 2 ) C 2 give rise to the same subvariety in the moduli space of curves which is the family ( ) of Table in [FGP ]. Therefore, we can try to decompose the Jacobian of C using this larger group.
/ Performing the calculation relative to this larger group we have (in the notation of table ): To conclude, notice that the nineth character of (C 4 × C 2 ) C 2 is not self dual, but we have to restrict it to Q 8 . Recalling that Q 8 (C 4 × C 2 ) C 2 , one sees that all the condition of Problem .
on page of [FH ] are fulfilled. Hence the restriction of this character to Q 8 is the only two dimensional irreducible representation, which is self dual. We remark that the L i must be CM elliptic curves, since H injects into M 2 (End(L i ) Q ).

Now recall that H⊗H
. This concludes the proof of theorem . .
. T h e c a s e K 2 = 8 a n d G = V 4 . -As an illustration of the proof of theorem . and in particular the computations performed by the MAGMA script, we will now study one example in detail. This example will occupy us for the next few pages.
Let E 1 and E 2 be elliptic curves, marked with two points {p i,1 , p i,2 } ⊂ E i . As group we take Then we define two G-covers f i : C i → E i , using the Riemann existence theorem (see e.g., [Mir , Sec. III]), by the following epimorphisms of groups: By construction {p i,1 , p i,2 } ⊂ E i is the branch locus, and above each p i,j , there are 2 ramification points with branching orders 2. Therefore, by the Riemann-Hurwitz formula the C i are curves of genus 3.
For a non trivial element g ∈ G, let χ g denote the nontrivial character of G, that annihilates g, let ρ g denote the character of G induced from the trivial character of the subgroup generated by g, let ρ 1 * be the regular character. We get We proceed by calculating the Hurwitz character ( . . ) relative to the first quotient (i = 1).
These results relate to row of table in the following way: [1, 1, 2] means that there is a 1-dimension reduced factor with multiplicity 1 corresponding to the second character in the MAGMA character table of V 4 , this is the character χ (0,1) . This is exactly the isogeny factor B χ (0,1) of J(C 1 ). Similarly [1, 1, 3] corresponds to the isogeny factor B χ (1,0) of J(C 2 ). Both J(C 1 ) and J(C 2 ) have an isogeny factor B χ (1,1) , which corresponds to [1,1,4] in table , and thus to the elliptic curves L 1 and L 2 .
Let us go further and build an algebraic K partner of S. To do that we consider the abelian surface A = P (ψ 1,(1,1) ) × P (ψ 2,(1,1) ) and divide modulo the natural involution. In this way we get a singular Kummer surface.
According to the propostion above and Shioda and Inose [SI ] the minimal resolution X of the singularities of Km(A) is a K surface whose transcendental part of H 2 (X) is isomorphic to the transcendental part of H 2 (A). Now consider the following diagram: All the morphisms in this diagram induce correspondences and by composing these correspondences we obtain an isomorphism h 2 (S) tra new → h 2 (X) tra that induces an isomorphism of Hodge structures H 2 new (S, Q) tra → H 2 (X, Q) tra .

Motivated K partners
Let S be a smooth projective complex surface with invariants p g (S) = q(S) = 2, and assume that the Albanese morphism α : S → A is surjective. In this section we attempt to answer question C of the introduction: Does there exist a K surface X together with an isomorphism ι : (H 2 new ) tra Q → H 2 (X, Q) tra that is motivated in the sense of Yves André?
. -In this section we prove the Tate and Mumford-Tate conjectures for surfaces that fall into type № = 3, 4, 6, 8, and 9 of table . We will use the language of motives, and specifically motivated cycles as introduced by André [And ].
This section is organised as follows: First we introduce notation and recall the definition of the Mumford-Tate group and the -adic monodromy groups. Then we will recall three conjectures that / are connected in the following sense: if two of the conjectures hold, then so does the third. These conjectures are (i) the Hodge conjecture; (ii) its -adic analogue known as the Tate conjecture; and (iii) the Mumford-Tate conjecture.
Starting from § . we recall the definition of motivated cycles in the sense of André [And ], and we quote the main theorems that describe the resulting category of motives. Once we we have all the machinery in place we turn our attention to the proof of the Tate and Mumford-Tate conjectures for the surfaces mentioned above.
. N o tat i o n . -Let K ⊂ C be a field, and let X be a smooth projective variety over K. We denote with H i (X) the singular cohomology group H i sing (X(C), Q). It is naturally endowed with a pure Hodge structure of weight i. Let be a prime number, and letK ⊂ C be the algebraic closure of K ∈ C. We denote with H i (X) the -adic étale cohomology group H i ét (XK, Q ). It is naturally endowed with a Galois representation Gal(K/K) → GL(H i (X)).
Artin's comparison theorem between étale cohomology and singular cohomology gives an isomorphism of vector spaces Recall from § . that we denote with CH * (X) the Chow ring of X with Q-coefficients. Recall the cycle class map cl : CH i (X C ) → H 2i (X)(i) for singular cohomology. There is also a cycle class map cl : CH i (X) → H 2i (X)(i) for étale cohomology. These are compatible with the comparison isomorphism H i (X) ∼ = H i (X) ⊗ Q ; we get the following commutative diagram: . M u m f o r d -Tat e g ro u p s . -Let V be a Q-Hodge structure. There is a representation of S = Res C/R G m on V ⊗ R: on complex points (z,z) acts on v ∈ V pq by v → z −pz−q v. (The minus signs are a historical convention.) Write h = h V for this representation S → GL(V ) R .
The Mumford-Tate group of V is the smallest algebraic subgroup G ⊂ GL(V ) over Q such that G R contains the image of h V . We denote the Mumford-Tate group of V with G MT (V ).
Alternatively, G MT (V ) may be defined using the Tannakian formalism. It is the algebraic group over Q associated with the Tannakian subcategory of Q-HS generated by V . If V is polarisable, then this subcategory generated by V is semisimple, and hence G MT (V ) is a reductive algebraic group.
Two more remarks are in place: First, observe that G MT (V ) is a connected algebraic group, since S is connected. Second, note that the subspace V ∩ V 0,0 C of Hodge classes in V is exactly the space of invariants V GMT(V ) .
-a d i c m o n o d ro m y g ro u p s . -Let K ⊂ C be a field of finite transcendence degree over Q. Let be a prime number, let V be a finite-dimensional Q -vector space, and let ρ : Gal(K/K) → GL(V ) be a representation that is continuous for the -adic topology on V .
The -adic monodromy group of V is the smallest algebraic subgroup G ⊂ GL(V ) over Q such that G(Q ) contains the image of ρ. We denote the -adic monodromy group of V with G (V ). In general, the algebraic group G (V ) is not connected; the identity component is denoted G • (V ).
An element of V is called a Tate class if it is invariant under an open subgroup of Gal(K/K).
In particular, the subspace of Tate classes in V is exactly the space of invariants V G • (V ) .
In general, the algebraic group G • (V ) is not reductive.
. C o n j e c t u r e ( H o d g e ) .
-Let X be a smooth projective variety over C. Then the image of cl : . C o n j e c t u r e ( Tat e ) . -Let X be a smooth projective variety over a number field K.
Then the image of cl : .
. R e m a r k . -To illustrate how these conjectures fit together, we make the following claims.
. If the Mumford-Tate conjecture is true for X, then the Hodge conjecture for X is equivalent to the Tate conjecture for X.
. If the Tate conjecture is true for all smooth projective varieties X over K, then the -adic monodromy groups are reductive. This follows from [Moo ].
. If the Hodge and Tate conjectures are true for all X, then the Mumford-Tate conjecture is true for all X.
. M o t i vat e d c y c l e s . -Let K be a subfield of C, and let X be a smooth projective variety over K. A class γ in H 2i (X) is called a motivated cycle of degree i if there exists an auxiliary smooth projective variety Y over K such that γ is of the form π * (α ∪ β), where π : X × Y → X is the projection, α and β are algebraic cycle classes in H * (X × Y ), and β is the image of β under the Hodge star operation. (Alternatively, one may use the Lefschetz star operation, see § of [And ].) Every algebraic cycle is motivated, and under the Lefschetz standard conjecture the converse holds as well. The set of motivated cycles naturally forms a graded Q-algebra. The category of motives over K, denoted Mot K , consists of objects (X, p, m), where X is a smooth projective variety over K, p is an idempotent motivated cycle on X × X, and m is an integer. A morphism (X, p, m) → (Y, q, n) is a motivated cycle γ of degree n − m on Y × X such that qγp = γ. We / denote with H mot (X) the object (X, ∆, 0), where ∆ is the class of the diagonal in X × X. The Künneth projectors π i are motivated cycles, and we denote with H i mot (X) the object (X, π i , 0). Observe that H mot (X) = i H i mot (X). This gives contravariant functors H mot (_) and H i mot (_) from the category of smooth projective varieties over K to Mot K . .
T h e o r e m . -The category Mot K is Tannakian over Q, semisimple, graded, and polarised. Every classical cohomology theory of smooth projective varieties over K factors via Mot K .
. D e f i n i t i o n . -Let K be a subfield of C. An abelian motive over K is an object of the Tannakian subcategory of Mot K generated by objects of the form H mot (X) where X is an abelian variety, or X = Spec(L) for some finite extension L/K, with L ⊂ C.
We denote the category of abelian motives over K with AbMot K .
. E x a m p l e . -If X/K is a K surface, then H 2 mot (X) is an abelian motive, by théorème . of [And ].
The Lefschetz motive Q(−1) is abelian, because any class of a hyperplane section in an abelian variety A will give a splitting H 2 mot (A) ∼ = M ⊕ Q(−1). .
T h e o r e m . -The Hodge realisation functor H : AbMot C → Q-HS is a full functor. Proof . See théorème . . of [And ]. .
T h e o r e m . -Let B be a reduced connected scheme of finite type over C. Let f : X → B be a smooth projective morphism, and ξ a global section of the sheaf R 2i f * Q(i). If there is a point 0 ∈ B(C) such that ξ 0 is motivated, then ξ b is motivated for all b ∈ B(C).
Proof . See théorème . of [And ]. . -We now have the notation and theory in place to answer question C (see § . ) about surfaces of general type with p g = q = 2. We give a partial answer to this question in the following results.

/ .
T h e o r e m . -Let α : S → B be a smooth projective family of surfaces of general type with invariants p g = q = 2 and dominant Albanese morphism. Assume that B is connected, and assume that there is one point 0 ∈ B such that the motive H 2 mot (S 0 ) of the fibre S 0 is an abelian motive. Then for every point b ∈ B, there exists a K surface X b , and an isomorphism of motives In particular, the motive H 2 mot (S b ) is abelian. Proof . The main idea of the proof is as follows: Using proposition . we construct a family X → B of Hodge-theoretic K partners. We then use theorem . to prove that H 2 mot (S 0 ) tra new is isomorphic to H 2 mot (X 0 ) tra new . Finally, this isomorphism spreads out to the other fibres via theorem . . We now make this sketch precise.
By replacing α : S → B with the pullback alongB → B, we may and do assume that B is simply connected. Let V denote the subvariation of Hodge structures of R 2 α * Q whose fibre V b at a point b ∈ B is H 2 (S b , Q) new . Fix a point b ∈ B. By proposition . we find that there is an , and a morphism of variations of Hodge structures ι : V| B • b → R 2 f * Q that is fibrewise a primitive embedding and a Hodge isometry on the transcendental lattices. We may view ι as a global section of the sheaf V ∨ | B • b ⊗ R 2 f * Q which is a subsheaf of R 4 (α, f ) * Q(2). Note that we may and do assume that b ∈ B • 0 ; indeed, if b / ∈ B • 0 , then we first prove the statement for all points in B • 0 , and then rerun the proof with a point 0 ∈ B • b ∩ B • 0 . Recall from example . that H 2 mot (X 0 ) is an abelian motive. Also note that H 2 mot (S 0 ) is abelian by assumption. Hence ι 0 is motivated, by theorem . . By theorem . , we see that ι b is motivated as well. This means that we obtain an isomorphism H 2 mot (S b ) tra new → H 2 mot (X b ) tra . In particular, the motive H 2 mot (S b ) tra new is abelian. To conclude that H 2 mot (S) is abelian, observe that H 2 mot (S) ∼ = H 2 mot (S) tra new ⊕ H 2 mot (S) tra old ⊕ Q(−1) r . The term H 2 mot (S) tra old is abelian, because it is the part coming from the Albanese surface, whose motive is abelian by definition. .
C o ro l l a ry . -Let K be a finitely generated subfield of C. Let S be a surface of general type over K with invariants p g = q = 2 and dominant Albanese morphism. Assume that S lies in one of the connected components of the Gieseker moduli space of surfaces of general type that contain a surface that is (semi-)isogenous to a product of curves. (That is, one of the types № , , , , or in  Proof . We first prove the Mumford-Tate conjecture for S. Let A be the Albanese variety of S. By theorem . there exists a K surface X such that H 2 mot (S) tra new ∼ = H 2 mot (X) tra . Possibly after replacing K by a finitely generated extension we may assume that X is defined over K. Hence the motive H mot (S) is an object in the Tannakian subcategory of Mot K generated by H mot (A) and H mot (X). Therefore it suffices to prove the Mumford-Tate conjecture for H mot (A) ⊕ H mot (X).
This follows from the main result of [Com ]. (See also [Vas ] and [Com ] for more general results on the Mumford-Tate conjecture for direct sums of abelian motives.) Recall that the Hodge conjecture is true for S, by the Lefschetz-(1, 1) theorem. Therefore the Tate conjecture for S is true, since it follows from the conjunction of the Hodge conjecture and the Mumford-Tate conjecture. Indeed, if γ ⊂ H 2 (S)(1) is a Tate class, then this means / that it is fixed by G • (S). We have just proven the Mumford-Tate conjecture for S, so we know that G • (S) ∼ = G MT (S) ⊗ Q . This means that γ ∈ H 2 (S)(1) ∼ = H 2 (S)(1) ⊗ Q is a Qlinear combination of G MT (S)-invariant classes in H 2 (S)(1). Those G MT (S)-invariant classes are precisely Hodge classes, and by the Lefschetz-(1, 1) theorem we know that they are in the image of the cycle class map. We conclude that γ is in the Q -span of the image of the -adic cycle class map.