American Journal of Mathematics

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

CONTENTS

    Matsushita, Daisuke.
  • Higher direct images of dualizing sheaves of Lagrangian fibrations
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    Abstract:
      Let f : XS 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 hp,q(S) = hp,q(n), where n = dim S. If Sn 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
    Graham, William.
  • Nonemptiness of symmetric degeneracy loci
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    Abstract:
      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 S2V* ⊗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
    Tataru, Daniel.
  • Rough solutions for the wave maps equation
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    Abstract:
      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
    Friedlander, Eric M.
    Pevtsova, Julia.
  • Representation-theoretic support spaces for finite group schemes
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    Abstract:
      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 ΨG: 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, ΨG 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
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    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 L1-Sectional curvature control
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    Abstract:
      Suppose that a sequence of Riemannian manifolds with Ricci curvature ≥ -k2 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 L1-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



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