65 research outputs found
Remarks on the abelian convexity theorem
This note contains some observations on abelian convexity theorems. Convexity
along an orbit is established in a very general setting using Kempf-Ness
functions. This is applied to give short proofs of the
Atiyah-Guillemin-Sternberg theorem and of abelian convexity for the gradient
map in the case of a real analytic submanifold of complex projective space.
Finally we give an application to the action on the probability measures.Comment: To appear on Proceedings of the American Mathematical Societ
Homogeneous bundles and the first eigenvalue of symmetric spaces
We prove the stability of the Gieseker point of an irreducible homogeneous
bundle over a rational homogeneous space. As an application we get a sharp
upper estimate for the first eigenvalue of the Laplacian of an arbitrary
Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.Comment: Some corrections suggested by the referee. To appear on Annales de
l'Institut Fourie
Stability of measures on K\"ahler manifolds
Let be a K\"ahler manifold and let be a compact group that
acts on in a Hamiltonian fashion. We study the action of on
probability measures on . First of all we identify an abstract setting for
the momentum mapping and give numerical criteria for stability, semi-stability
and polystability. Next we apply this setting to the action of
on measures. We get various stability criteria for measures on K\"ahler
manifolds. The same circle of ideas gives a very general surjectivity result
for a map originally studied by Hersch and Bourguignon-Li-Yau.Comment: Final version. To appear on Advances in Mathematic
Shimura varieties in the Torelli locus via Galois coverings
Given a family of Galois coverings of the projective line we give a simple
sufficient condition ensuring that the closure of the image of the family via
the period mapping is a special (or Shimura) subvariety in A_g. By a computer
program we get the list of all families in genus up to 8 satisfying our
condition. There is no family in genus 8, all of them are in genus at most 7.
These examples are related to a conjecture of Oort. Among them we get the
cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot,
Rohde, Moonen and others) and the abelian non-cyclic examples found by
Moonen-Oort. We get 7 new non-abelian examples.Comment: Final version. To appear on Intenational Mathematics Research Notice
Polar orbitopes
We study polar orbitopes, i.e. convex hulls of orbits of a polar
representation of a compact Lie group. The face structure is studied by means
of the gradient momentum map and it is shown that every face is exposed and is
again a polar orbitope. Up to conjugation the faces are completely determined
by the momentum polytope. There is a tight relation with parabolic subgroups:
the set of extreme points of a face is the closed orbit of a parabolic subgroup
of G and for any parabolic subgroup the closed orbit is of this form.Comment: 24 pages. To appear on Communications in Analysis and Geometr
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