24 research outputs found
On Sasaki-Einstein manifolds in dimension five
We prove the existence of Sasaki-Einstein metrics on certain simply connected
5-manifolds where until now existence was unknown. All of these manifolds have
non-trivial torsion classes. On several of these we show that there are a
countable infinity of deformation classes of Sasaki-Einstein structures.Comment: 18 pages, Exposition was expanded and a reference adde
Equidistribution of zeros of holomorphic sections in the non compact setting
We consider N-tensor powers of a positive Hermitian line bundle L over a
non-compact complex manifold X. In the compact case, B. Shiffman and S.
Zelditch proved that the zeros of random sections become asymptotically
uniformly distributed with respect to the natural measure coming from the
curvature of L, as N tends to infinity. Under certain boundedness assumptions
on the curvature of the canonical line bundle of X and on the Chern form of L
we prove a non-compact version of this result. We give various applications,
including the limiting distribution of zeros of cusp forms with respect to the
principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the
higher dimensional case of arithmetic quotients and the case of orthogonal
polynomials with weights at infinity. We also give estimates for the speed of
convergence of the currents of integration on the zero-divisors.Comment: 25 pages; v.2 is a final update to agree with the published pape
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
Episcience : une plate-forme multidisciplinaire pour les Ă©pirevues
Nous présenterons les caractéristiques de la plate-forme Episciences hébergée par le CCSD sous la triple tutelle du CNRS, de l\u27INRIA et de l\u27Université de Lyon. Cette plate-forme entiÚrement automatisée de revues en open access fonctionne en "sur-couche" au dessus des archives ouvertes telles que HAL et arXiv
Correnti positive: uno strumento per l'analisi globale su varietĂ complesse
This paper is a survey on positive currents on complex manifolds. It is essentially divided into two parts. In the first part the author
illustrates principal results on closed, positive currents: This is a very important class of currents
for complex manifolds because they are a natural generalization of submanifolds. In the second
part pluriharmonic and plurisubharmonic positive currents are illustrated; this class is important
because of the characterization of compact KĂ€hler manifolds in terms of currents by Harvey and
Lawson. At the end of the paper an appendix, containing some preliminaries on the subject, helps
the nonexpert reader