315 research outputs found
K-area, Hofer metric and geometry of conjugacy classes in Lie groups
Given a closed symplectic manifold we introduce a certain
quantity associated to a tuple of conjugacy classes in the universal cover of
the group by means of the Hofer metric on
. We use pseudo-holomorphic curves involved in the
definition of the multiplicative structure on the Floer cohomology of a
symplectic manifold to estimate this quantity in terms of actions
of some periodic orbits of related Hamiltonian flows. As a corollary we get a
new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of
products of unitary matrices. As another corollary we get a new proof of the
geodesic property (with respect to the Hofer metric) of Hamiltonian flows
generated by certain autonomous Hamiltonians. Our main technical tool is K-area
defined for Hamiltonian fibrations over a surface with boundary in the spirit
of L.Polterovich's work on Hamiltonian fibrations over .Comment: Corrected final version, accepted for publication in Inventiones
Mathematica
Positivity of relative canonical bundles and applications
Given a family of canonically polarized manifolds, the
unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the
relative canonical bundle . We use a global elliptic
equation to show that this metric is strictly positive on , unless
the family is infinitesimally trivial.
For degenerating families we show that the curvature form on the total space
can be extended as a (semi-)positive closed current. By fiber integration it
follows that the generalized Weil-Petersson form on the base possesses an
extension as a positive current. We prove an extension theorem for hermitian
line bundles, whose curvature forms have this property. This theorem can be
applied to a determinant line bundle associated to the relative canonical
bundle on the total space. As an application the quasi-projectivity of the
moduli space of canonically polarized varieties
follows.
The direct images , , carry natural hermitian metrics. We prove an
explicit formula for the curvature tensor of these direct images. We apply it
to the morphisms that are induced by the Kodaira-Spencer map and obtain a differential
geometric proof for hyperbolicity properties of .Comment: Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in
Invent. mat
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