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 …
Homotopy groups of pullbacks of varieties
In [2, § 9] there is a general result of Fulton and Lazarsfeld relating the homotopy groups of a subvariety of in a certain range of dimensions with those of its pullback under a holomorphic map in the corresponding range of dimensions. It is asked in [2, § 10] whether here is a corresponding result with replaced by a general rational homogeneous manifold, Y, and with the range of dimensions alluded to above shifted by the ampleness of the holomorphic tangent bundle of Y in the sense of [4]. In this paper we use the techniques of [4, 5, 6, 7] to answer this question in the affirmative.
The minimal number of singular fibers of a semistable curves over P^1
In this paper, we shall prove Beauville's conjecture: if $f:S \to P^1$ is a non-trivial semistable fibration of genus g>1, then $f$ admits at least 5 singular fibers. We have also constructed an example of genus 2 with 5 singular fibers. This paper will appear in the Journal of Algebraic Geometry.