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 …
Fibred surfaces, varieties isogenous to a product and related moduli spaces
A fibration of an algebraic surface S over a curve B, with fibres of genus at least 2, has constant moduli iff it is birational to the quotient of a product of curves by the action of a finite group G. A variety isogenous to a (higher) product is the quotient of a product of curves of genus at least 2 by the free action of a finite group. Theorem B gives a characterization of surfaces isogenous to a higher product in terms of the fundamental group and of the Euler number. Theorem C classifies the groups thus occurring and shows that, after fixing the group and the Euler number, one obtains an irreducible moduli space. The result of Theorem B is extended to higher dimension in Theorem G, thus generalizing (cf. also Theorem H) results of Jost-Yau and Mok concerning varieties whose universal cover is a polydisk. Theorem A shows that fibrations where the fibre genus and the genus of the base B are at least 2 are invariants of the oriented differentiable structure. The main Theorems D and E characterize surfaces carrying constant moduli fibrations as surfaces having a Zariski open set satisfying certain topological conditions (e.g., having the right Euler number, the right fundamental group and the right fundamental group at infinity).
HYPERGOLIC REACTION MECHANISMS OF CATALYTICALLY PROMOTED FUELS WITH ROCKET GRADE HYDROGEN PEROXIDE
Abstract The coupling of the physical and chemical processes controlling the ignition delays (τ ign ) of hypergolic propellants complicates the direct analysis of the transient ignition processes. The transport properties of both the propellants and the ambient gas, and the heat of vaporization and reaction affect τ ign and must be considered experimentally. Furthermore, since hypergolic ignition occurs, by definition, on contact, the mechanics of initial contact and subsequent mixing can affect τ ign and must also be considered. Experimental results from a study of the hypergolic ignition of a catalytically promoted fuel with rocket grade hydrogen peroxide (RGHP) are presented. Statistical analysis of the experimental data was conducted to evaluate the relative effects of RGHP concentration, initial ambient pressure, and the ambient gas on τ ign . At RGHP concentrations of less than 92% in water, stable ignition was difficult to achieve. At sufficiently high Lewis number, an oscillatory ignition process is observed. A phenomenological model comprising mixing, RGHP decomposition, fuel heating and vaporization, and auto-ignition is presented and compared to the experimental results.