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L2 harmonic forms for a class of complete Kahler metrics

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journal contribution
posted on 31.03.2015, 10:02 by Eugenie Hunsicker
The 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 Journal

Volume

50

Issue

2

Pages

339 - 349

Citation

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 Michigan

Version

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

2002

Notes

This article was published in the Michigan Mathematical Journal and is available here with the kind permission of the publisher.

ISSN

0026-2285

Language

en

Exports

Loughborough Publications

Keywords

Exports