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 …
Mathematics and Statistics The geometry of second symmetric products of curves
This paper is a short summary of the main results in the thesis [1] and in the paper [40]. We deal throughout with several problems on the surfaces obtained as second symmetric products of smooth projective curves. In particular, we treat both some attempts at extending the notion of gonality for curves and some classical problems, as the study of the ample cone in the Neron-Severi group. Moreover, we develop a family of examples of Lagrangian surfaces having particular topological properties.
Birth-death processes with killing
The purpose of this note is to point out that Karlin and McGregor's integral representation for the transition probabilities of a birth-death process on a semi-infinite lattice with an absorbing bottom state remains valid if one allows the possibility of absorption into the bottom state from any other state. Conditions for uniqueness of the minimal transition function are also given.