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 …
Density of rational points on elliptic K3 surfaces
Let $X$ be a K3 surface defined over a number field $K$. Assume that $X$ admits a structure of an elliptic fibration or an infinite group of automorphisms. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points is Zariski dense in $X$.
State estimation for integrated vehicle dynamics control
This paper discusses a vehicle controller and a state estimator that was implemented and tested in a vehicle equipped with a combined braking and chassis control system to improve handling. The vehicle dynamics controller consists of a feed forward body roll compensation and a feedback stability controller. The estimator is based on a non-linear vehicle model combined with an extended Kalman filter. The Kalman filter is also connected to an algorithm, which estimates the tyre-road friction coefficient. The developed estimator was tested in a vehicle. Good results were achieved for normal and extreme driving manoeuvers.