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

Abstract

In this paper we study the cohomology of smooth projective complex surfaces S of general type with invariants pg = 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 K surface X that we call the K partner of S. Furthermore, we show that in suitable cases we can geometrically construct the K partner X and an algebraic correspondence in S ×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.

A note on a family of surfaces with $$p_g=q=2$$ and $$K^2=7$$

“…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. We can make the following general observations (see also [CoPe20]). It holds:…”

“…We illustrate only the demonstration strategy in the realm of motives. The main idea in [CoPe20] is that for surfaces S with p g = 2 it is sometimes possible to decompose the weight 2 Hodge structure into two Hodge substructures of K3 type and see that these Hodge substructures are indeed the Hodge structures of either Abelian surfaces or K3 surfaces which are (birational) quotients of S. This geometric construction makes possible to consider the theory of motivated cycles introduced by André, and to decompose the motive of S into two abelian motives of K3 type. For these motives the Mumford-Tate conjecture is known.…”

“…For these motives the Mumford-Tate conjecture is known. This, together with the main results of [Com16] and [Com19] allows to prove the Mumford-Tate conjecture for S. In [CoPe20] it was used the fact that the surfaces studied are of maximal Albanese dimension hence there is naturally an Abelian surface as a quotient surface.…”

“…These surfaces give rise to three disjoint open subsets in the Gieseker moduli space M can 2, 2, 7 which are all irreducible, generically smooth of dimension 2, that we shall denote by M 1 , M 2 and M 4 .The Mumford-Tate conjecture for surfaces is still an open problem and only in very few examples it has been verified as true, see for example [Moo17a]. A strategy to prove the conjecture for surfaces with p g = q = 2 and of maximal Albanese dimension is outlined in the article [CoPe20]. The strategy reduces to finding geometric quotients X of S that are K3 surfaces whose weight 2 Hodge structure is a sub-Hodge structure of the weight 2 Hodge structure of S orthogonal to the the sub-Hodge structure coming from the Albanese surface of S. This strategy proved to be very successful in many cases, see [CoPe20].…”

“…A strategy to prove the conjecture for surfaces with p g = q = 2 and of maximal Albanese dimension is outlined in the article [CoPe20]. The strategy reduces to finding geometric quotients X of S that are K3 surfaces whose weight 2 Hodge structure is a sub-Hodge structure of the weight 2 Hodge structure of S orthogonal to the the sub-Hodge structure coming from the Albanese surface of S. This strategy proved to be very successful in many cases, see [CoPe20]. The first question toward exploiting the strategy is to calculate the automorphism of the surfaces S and then classify all possible quotients.…”