2 research outputs found
The subelliptic heat kernel on SU(2): Representations, Asymptotics and Gradient bounds
The Lie group SU(2) endowed with its canonical subriemannian structure
appears as a three-dimensional model of a positively curved subelliptic space.
The goal of this work is to study the subelliptic heat kernel on it and some
related functional inequalities.Comment: Update: Added section + Correction of typo
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