ALGEBRAIC SURFACES WITH AUTOMORPHISMS

In my thesis I worked on two different projects, both related with projective surfaces with automorphisms. In the first one I studied Abelian surfaces with an automorphism and quaternionic multiplication: this work has already been accepted for publication in the Canadian Journal of Mathematics. In the second project I treat surfaces isogenous to a product of curves and their cohomology. Abelian Surfaces with an Automorphism The Abelian surfaces, with a polarization of a fixed type, whose endomorphism ring is an order in a quaternion algebra are parametrized by a curve, called Shimura curve, in the moduli space of polarized Abelian surfaces. There have been several attempts to find concrete examples of such Shimura curves and of the Abelian surfaces over this curve. In [HM95] Hashimoto and Murabayashi find Shimura curves as the intersection, in the moduli space of principally polarized Abelian surfaces, of Humbert surfaces. Such Humbert surfaces are now known “explicitly” in many other cases and this might allow one to find explicit models of other Shimura curves. Other approaches are taken in [Elk08] and [PS11]. We consider the rather special case where one of the Abelian surfaces in the family is the selfproduct of an elliptic curve. We assume this elliptic curve to have an automorphism of order three or four. For a fixed product polarization of type (1, d), we denote by Hj,d the set of the deformations of the selfproduct with the automorphism of order j. We prove the following theorem: Theorem. Let j ∈ {3,4}, d ∈ Z, d > 0 and let τ ∈ Hj,d, so that the Abelian surface Aτ,d has an automorphism φj of order j. Then the endomorphism algebra of Aτ,d also contains an element ψj with ψj^2 = d. Moreover, for a general τ ∈ Hj,d one has End(Aτ,d)Q =(−j,d)/Q where (a, b)/Q := Q1 ⊕ Qi ⊕ Qj ⊕ Qk is the quaternion algebra with i^2 = a, j^2 =b and ij=−ji. It is easy compute for which d the quaternion algebra (−j,d)/Q is a skew field: for these d the general Abelian surface in the family Hj,d is simple. In particular this provides examples of simple Abelian surfaces with an automorphism of order three and four. This construction, together with well-known results about automorphisms of Abelian surfaces (see [BL04, Chapter 13]), leads to: Theorem. Let A be a simple Abelian surface and φ ∈ Aut(A) a non-trivial automorphism of finite order. Then ord(φ) ∈ {3, 4, 5, 6, 10}. We focus in particular on the family H3,2 of Abelian surfaces with an automorphism of order three and a polarization of type (1,2). In [Bar87] Barth provides a description of a moduli space M2,4, embedded in P5, of (2, 4)-polarized Abelian surfaces with a level structure. Since the polarized Abelian surfaces we consider have an automorphism of order three, the corresponding points in M2,4 are fixed by an automorphism of order three of P^5. This allows us to explicitly identify the Shimura curve in M2,4 that parametrizes the Abelian surfaces with quaternionic multiplication by the maximal order O6 in the quaternion algebra with discriminant 6. It is embedded as a line in M2,4 ⊂ P5 and the symmetric group S4 acts on this line by changing the level structures. According to Rotger [Rot04], an Abelian surface with endomorphism ring O6 is the Jacobian of a unique genus two curve. We show explicitly how to find such genus two curves, or rather their images in the Kummer surface embedded in P5 with a (2,4)-polarization. These curves were already been considered by Hashimoto and Murabayashi in [HM95]: we give the explicit relation between their description and ours. Moreover we find a Humbert surface in M2,4 that parametrizes Abelian surfaces with Z( 2) in the endomorphism ring. Cohomology of surfaces isogenous to a product Surfaces isogenous to a product of curves provide examples of surfaces of general type with many different geometrical invariants. They have been introduced by Catanese in [Cat00]: Definition. A smooth surface S is said to be isogenous to a product (of curves) if it is isomorphic to a quotient (C×D)/G where C and D are curves of genus at least one and G is a finite group acting freely on C × D. We say that a surface isogenous to a product is of mixed type if there exists a element of G interchanging the two curves; otherwise, if G acts diagonally on the product, we say that the surface is of unmixed type. A surface isogenous to a product is of general type if the genus of both curves, C and D, is greater or equal to 2: in this case we say that the surface is isogenous to a higher product. The cohomology groups of a surface S = (C × D)/G isogenous to a product of unmixed type are determined by the action of the group G on the cohomology groups of the curves. Moreover the action of an automorphism group G on a smooth curve C forces a decomposition of the first cohomology group, as described in [BL04, section 13.6] and in [Roj07]: Proposition (Group algebra decomposition). Let G be a finite group acting on a curve C. Let W1, ..., Wr denote the irreducible rational representations of G and let ni := dimDi (Wi), with Di := EndG(Wi), for i = 1, ..., r. Then there are rational Hodge substructures B1, ..., Br such that H1(C, Q) ≃ n1B1+...+nrBr. From this a decomposition of the cohomology groups of the surface S follows directly. We apply these results to surfaces isogenous to a higher product of unmixed type with χ(OS) = 2 and q(S) = 0: they have been studied and classified by Gleissner in [Gle13]. For these surfaces the Hodge diamond is fixed and in particular the Hodge numbers of the second coho- mology groups are the same as those of an Abelian surface. From Gleissner’s classification we obtain a complete list of the 21 possible groups G. We proved that the second cohomology group of these surfaces can be described explicitly as follows: Theorem. Let S be a surface isogenous to a higher product of unmixed type with χ(OS) = 2, q(S) = 0. Then there exist two elliptic curves E1 and E2 such that H2(S, Q) ∼= H2(E1 × E2, Q) as rational Hodge structures. In general it is not possible to construct these elliptic curves “geometrically” using the action of G. More precisely there are no intermediate coverings πF : C → C/F and πH : D → D/H, F, H ≤ G with C/F = E1 and D/H = E2: we can only prove that such elliptic curves must exist. The proof of the theorem is standard for all but four groups: in these cases we study one by one the corresponding surfaces in order to construct the elliptic curves. As a further application we use this approach to study some surfaces isogenous to a higher product with pg = q = 2, in particular those are of Albanese general type.