Volume 2010, Issue 644

We define an integral Borel-Moore homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology. Both types of groups are expected to be finitely generated, and sit in a long exact sequence with higher Chow groups of zero-cycles.

By analogy with Thompson's classification of nonsolvable finite N -groups, we classify groups of finite Morley rank with solvable local subgroups of even and of mixed type. We also consider several aspects reminiscent of “small” groups of finite Morley rank of odd type.

In this paper we prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubinstein but to our knowledge its proof has never been published. Our proof follows ideas of Simon and uses an extension of a famous result of Meeks, Simon and Yau on the convergence of minimizing sequences of isotopic surfaces. This result is proved in the second part of the paper.

We study a particular class of rationally connected manifolds, , such that two general points x , x ′ ∈ X may be joined by a conic contained in X . We prove that these manifolds are Fano, with b 2 ≦ 2. Moreover, a precise classification is obtained for b 2 = 2. Complete intersections of high dimension with respect to their multi-degree provide examples for the case b 2 = 1. The proof of the classification result uses a general characterization of rationality, in terms of suitable covering families of rational curves.

Let A be a quotient of J 0 ( N ) associated to a newform ƒ such that the special L -value of A (at s = 1) is non-zero. We give a formula for the ratio of the special L -value to the real period of A that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of ƒ with eigenforms of positive analytic rank. We use the techniques of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime q divides this factor, then q divides either the order of the Shafarevich-Tate group or the order of a component group of A . Suppose p is an odd prime such that p 2 does not divide N, p does not divide the order of the rational torsion subgroup of A , and ƒ is congruent modulo a prime ideal over p to an eigenform whose associated abelian variety has positive Mordell-Weil rank. Then we show that p divides the factor mentioned above; in particular, p divides the numerator of the ratio of the special L -value to the real period of A . Both of these results are as implied by the second part of the Birch and Swinnerton-Dyer conjecture, and thus provide theoretical evidence towards the conjecture.

Let Y be a normal projective variety and π : X → Y a projective holomorphic symplectic resolution. Namikawa proved that the Kuranishi deformation spaces Def ( X ) and Def ( Y ) are both smooth, of the same dimension, and π induces a finite branched cover ƒ : Def ( X ) → Def ( Y ). We prove that ƒ is Galois. We proceed to calculate the Galois group G , when X is simply connected, and its holomorphic symplectic structure is unique, up to a scalar factor. The singularity of Y is generically of ADE-type, along every codimension 2 irreducible component B of the singular locus, by Namikawa's work. The modular Galois group G is the product of Weyl groups of finite type, indexed by such irreducible components B . Each Weyl group factor W B is that of a Dynkin diagram, obtained as a quotient of the Dynkin diagram of the singularity-type of B , by a group of Dynkin diagram automorphisms. Finally we consider generalizations of the above set-up, where Y is affine symplectic, or a Calabi-Yau threefold with a curve of ADE-singularities. We prove that ƒ : Def ( X ) → Def ( Y ) is a Galois cover of its image. This explains the analogy between the above results and related work of Nakajima, on quiver varieties, and of Szendrői on enhanced gauge symmetries for Calabi-Yau threefolds.

Let be open. We show that each homeomorphism satisfies . If we moreover assume that ƒ has finite distortion, then and ƒ –1 has finite distortion. The main ingredient is a new result on change of variables in integral (area and coarea formula) for such mappings.