American Journal of Mathematics, Volume 127, 2005 - Table of Contents

American Journal of Mathematics
Volume 127, Number 2, April 2005

CONTENTS

Matsushita, Daisuke.

Higher direct images of dualizing sheaves of Lagrangian fibrations [Access Article in PDF] Abstract:

Let f : X → S be a Lagrangian fibration between projective varieties. We prove that
if S is smooth. Suppose that X is an irreducible symplectic manifold or a certain
moduli space of semistable torsion free sheaves on a K3 surface, the Hodge numbers satisfy h^{p,q}(S) =
h^{p,q}(^{n}), where n = dim S. If S ≅ ^{n} and X is an irreducible symplectic manifold, there exists a
hypersurface of the Kuranishi space of X such that every member of the Kuranishi family over
admits a Lagrangian fibration over ^{n}.
(pages 243-259.)
Abstract in Tex

Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume
that V is equipped with a quadratic form with values in a line bundle L and that S^{2}V* ⊗L is ample.
Suppose that the maximum rank of the quadratic form at any point of X is r > 0. The main result
of this paper is that if d > N - r, the locus of points where the rank of the quadratic form is at
most r - 1 is nonempty. We give some applications to subschemes of matrices, and to degeneracy
loci associated to embeddings in projective space. The paper concludes with an appendix on Gysin
maps. The main result of the appendix, which may be of independent interest, identifies a Gysin
map with the natural map from ordinary to relative cohomology.
(pages 261-292.)
Abstract in Tex

We consider the wave maps equation with values into a Riemannian manifold which is
isometrically embedded in ^{m}. Our main result asserts that the Cauchy problem is globally well-posed
for initial data which is small in the critical Sobolev spaces. This extends and completes
recent work of Tao and other authors.
(pages 293-377.)
Abstract in Tex

We introduce the space P(G) of abelian p-points of a finite group scheme over an algebraically
closed field of characteristic p > 0. We construct a homeomorphism Ψ_{G}: P(G) → Proj |G| from P(G) to the projectivization of the cohomology variety for any finite group G. For an elementary
abelian p-group (respectively, an infinitesimal group scheme), P(G) can be identified with the
projectivization of the variety of cyclic shifted subgroups (resp., variety of 1-parameter subgroups).
For a finite dimensional G-module M, Ψ_{G} restricts to a homeomorphism P(G)_{M} → Proj |G|_{M},
thereby giving a representation-theoretic interpretation of the cohomological support variety.
(pages 379-420.)
Abstract in Tex

van Diejen, J. F.

Scattering theory of discrete (pseudo) Laplacians on a Weyl chamber [Access Article in PDF] Abstract:

To a crystallographic root system we associate a system of multivariate orthogonal polynomials diagonalizing an integrable system of discrete pseudo Laplacians on the Weyl chamber. We develop the time-dependent scattering theory for these discrete pseudo Laplacians and determine the corresponding wave operators and scattering operators in closed form. As an application, we describe the scattering behavior of certain hyperbolic Ruijsenaars-Schneider type lattice Calogero-Moser models associated with the Macdonald polynomials.
(pages 421-458.)
Abstract in Tex

Sprouse, Chadwick.

Manifolds with lower Ricci and L^{1}-Sectional curvature control [Access Article in PDF] Abstract:

Suppose that a sequence of Riemannian manifolds with Ricci curvature ≥ -k^{2} converges
to a Gromov-Hausdorff limit X. We show that if the amount of sectional curvature below K of the
limiting manifolds approaches 0 in a suitable L^{1}-sense, then X is an Alexandrov space of curvature
≥ K. As applications we present several generalizations of classical theorems.
(pages 459-469.)
Abstract in Tex