Lagrangian subvarieties of Abelian fourfolds
Let (W, ω) be a smooth projective algebraic variety of dimension 2n over C together with a holomorphic (2, 0)-form of maximal rank 2n. A subvariety X ⊂ W is called weakly lagrangian if dim X ≤ n and if the restriction of ω to X is trivial (notice that X can be singular). An n-dimensional subvariety X ⊂ W with this property is called lagrangian. For example, any curve C contained in a K3 or abelian surface S is lagrangian. Further examples of lagrangian subvarieties are obtained by taking a curve C ⊂ S and by considering the corresponding symmetric products. Alternatively, one could look at a product of different curves (of genus > 1) inside a product of abelian varieties. We will say that a variety X ⊂ W is fibered if it admits a dominant map onto a curve of genus > 1. In this note we construct examples of nonfibered lagrangian surfaces in abelian varieties. Our motivation comes from the following Problem 1.1. — Find examples of projective surfaces with a nontrivial fundamental group. In particular, find examples where the fundamental group has a nontrivial nilpotent tower. If X is fibered over a curve C of genus > 1 then the fundamental group π 1 (X) surjects onto a subgroup of finite index in π 1 (C) and consequently both π 1 (X) and its nilpotent tower are big. Therefore, we are interested in examples of surfaces where the nontriviality of π 1 (X) is not induced from curves. Consider the map to the Albanese variety alb : X → Alb(X). One is interested in situations where the natural map H 2 (Alb(X), C) → H 2 (X, C) has a nontrivial kernel – the triviality of the kernel implies the triviality of the nilpotent tower (tensor Q). Such examples were given by Campana ([3], Cor. 1.2) and Sommese-Van de Ven ([7]). However, there the kernel LAGRANGIAN SUBVARIETIES OF ABELIAN FOURFOLDS 3 was found in the map Pic(Alb(X)) → Pic(X). In their construction the fundamental group of the variety is a central extension of an abelian group (and the lower central series has only two steps). The lagrangian property of alb(X) ⊂ Alb(X) implies that there is a nontrivial kernel in H 2,0 (rather than on the level of the Picard groups). We produce an infinite series of surfaces X of different topological types which are …
Construction of Nikulin configurations on some Kummer surfaces and applications
A Nikulin configuration is the data of 16 disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration $$\mathcal {C}$$C, then X is a Kummer surface $$X=\mathrm{Km}(B)$$X=Km(B) where B is an Abelian surface determined by $$\mathcal {C}$$C. Let B be a generic Abelian surface having a polarization M with $$M^{2}=k(k+1)$$M2=k(k+1) (for $$k>0$$k>0 an integer) and let $$X=\mathrm{Km}(B)$$X=Km(B) be the associated Kummer surface. To the natural Nikulin configuration $$\mathcal {C}$$C on $$X=\mathrm{Km}(B)$$X=Km(B), we associate another Nikulin configuration $$\mathcal {C}'$$C′; we denote by $$B'$$B′ the Abelian surface associated to $$\mathcal {C}'$$C′, so that we have also $$X=\mathrm{Km}(B')$$X=Km(B′). For $$k\ge 2$$k≥2 we prove that B and $$B'$$B′ are not isomorphic. We then construct an infinite order automorphism of the Kummer surface X that occurs naturally from our situation. Associated to the two Nikulin configurations $$\mathcal {C},$$C,$$\mathcal {C}'$$C′, there exists a natural bi-double cover $$S\rightarrow X$$S→X, which is a surface of general type. We study this surface which is a Lagrangian surface in the sense of Bogomolov-Tschinkel, and for $$k=2$$k=2 is a Schoen surface.
Polynomial bounds for the number of automorphisms of a surface of general type
— We study Weierstra(3 points and discriminants on algebraic surfaces of general type and we give applications to a polynomial type estimate in c^ for the order of the automorphism group.