Volume 2012, Issue 664

The variant A3w of Ma, Trudinger and Wang's condition for regularity of optimal transportation maps is implied by the non-negativity of a pseudo-Riemannian curvature—which we call cross-curvature —induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature non-negativity is preserved for products of two manifolds; (2) both A3w and cross-curvature non-negativity are inherited by Riemannian submersions, as is domain convexity for the exponential maps; and (3) the n -dimensional round sphere satisfies cross-curvature non-negativity. From these results, a large new class of Riemannian manifolds satisfying cross-curvature non-negativity (thus A3w) is obtained, including many whose sectional curvature is far from constant. All known obstructions to the regularity of optimal maps are absent from these manifolds, making them a class for which it is natural to conjecture that regularity holds. This conjecture is confirmed for certain Riemannian submersions of the sphere such as the complex projective spaces ℂℙ n .

We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul-Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split off, locally, a sphere or a compact Lie group with a bi-invariant metric. From this it follows easily how to obtain the full isometry group of a naturally reductive space. This generalizes known classification results of Onishchick, for normal homogeneous spaces with simple group of isometries, and Shankar, for homogeneous spaces of positive curvature. This also answers a question posed by J. Wolf and Wang–Ziller. Namely, to explain why the presentation group of an isotropy irreducible space, strongly or not, cannot be enlarged (except for spheres, or for compact simple Lie groups with a bi-invariant metric).

In this paper, we study criteria for the equality between the homological and the cohomological dimension of a group. In particular, we obtain criteria for the freeness of finitely generated groups of homological dimension one. To that end, we examine two types of conditions under which flat modules are projective over a ring R .

Let F be a global field, let S ∞ be the set of archimedean primes of F and let S be any nonempty finite set of primes of F containing S ∞ . In this paper we study the Néron S -class group C A ,  F ,  S of an abelian variety A defined over F . In the well-known analogy that exists between the Birch and Swinnerton–Dyer conjecture for A over F and the analytic class number formula for the field F (in the number field case), the finite group C A ,  F ,  S ∞ (not the Tate–Shafarevich group of A ) is a natural analog of the ideal class group of F .

We consider multiparameter dynamics on the space of unimolular lattices. Along with quantitative nondivergence we prove that multiplicative Diophantine exponents of hyperplanes are inherited by their nondegenerate submanifolds.

We study the existence of rational points on modular curves of 𝒟-elliptic sheaves over local fields and the structure of special fibres of these curves. We discuss some applications which include finding presentations for arithmetic groups arising from quaternion algebras, finding the equations of modular curves of 𝒟-elliptic sheaves, and constructing curves violating the Hasse principle.

In this paper we give a splitting criterion for uniform vector bundles on Fano manifolds covered by lines. As a consequence, we classify low rank uniform vector bundles on Hermitian symmetric spaces and Fano bundles of rank two on Grassmannians.

Using explicit identifications between algebraic and analytic theta functions, we compare the algebraic constructions of toroidal compactifications by Faltings–Chai and the author, with the analytic constructions of toroidal compactifications following Ash–Mumford–Rapoport–Tai. As one of the applications, we obtain the corresponding comparison for Fourier–Jacobi expansions of holomorphic automorphic forms.

We show (a) that any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ℂ n with the Euclidean metric is flat; (b) that any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ℂ n with the pseudo-Euclidean metric is flat if the Hessian of the potential is bounded below quadratically; and (c) the Hermitian counterpart of (b) for the Kähler Ricci flow.