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.
A new class of numerical sequences and its applications to sine and cosine series
First, it is proved that the class of classical quasi-monotone sequences is not comparable to the newly defined class of sequences of ‘rest bounded variation’. Considering this result, we prove three sample theorems for sequences of rest bounded variation, being analogues of the theorems proved earlier for monotone or quasi-monotone sequences. One of them gives a partial answer to a question raised by R. P. Boas Jr.