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Download fileL2 harmonic forms for a class of complete Kahler metrics
journal contribution
posted on 2015-03-31, 10:02 authored by Eugenie HunsickerThe Hodge theorem for compact manifolds states that every real cohomology class
of a compact manifold M is represented by a unique harmonic form. That is, the
space of solutions to the differential equation .d Cd
/ D 0 on L2 forms over M;
a space that depends on the metric on M; is canonically isomorphic to the purely
topological real cohomology space of M: This isomorphism is enormously useful
because it provides a way to transform theorems from geometry into theorems in
topology and vice versa. No such result holds in general for complete noncompact
manifolds, but in many specific cases there are Hodge-type theorems. One of the
oldest is the description, due to Atiyah, Patodi, and Singer [1], of the space of L2
harmonic forms on a manifold with complete cylindrical ends. By calculating the
solutions to the equation for harmonic forms on the cylindrical ends, they showed
that the space of L2 harmonic forms is isomorphic to the image of the relative cohomology
of the manifold in the absolute cohomology. Another Hodge-type result
was found by Zucker [14] for a natural class of metrics called Poincaré metrics.
These metrics, first constructed by Cornalba and Griffiths [4], are complete Kähler
metrics with hyperbolic cusp-type singularities at isolated points on a Riemann
surface. Zucker showed that the space of L2 forms on a Riemann surface that are
harmonic with respect one of these metrics is isomorphic to the standard cohomology
of the surface. This result was extended by Cattani, Kaplan, and Schmid [3]
to analogous metrics on bundles over projective varieties with singularities along
a divisor. These metrics can be thought of as complete Kähler metrics on the noncompact
manifold given by removing the divisor.
History
School
- Science
Department
- Mathematical Sciences
Published in
Michigan Mathematical JournalVolume
50Issue
2Pages
339 - 349Citation
HUNSICKER, E., 2002. L2 harmonic forms for a class of complete Kahler metrics. Michigan Mathematical Journal, 50 (2), pp. 339 - 349.Publisher
Mathematics Department, University of MichiganVersion
- VoR (Version of Record)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Publication date
2002Notes
This article was published in the Michigan Mathematical Journal and is available here with the kind permission of the publisher.ISSN
0026-2285Publisher version
Language
- en