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Annals of Global Analysis and Geometry

, Volume 48, Issue 1, pp 75–85 | Cite as

Einstein metrics, harmonic forms, and symplectic four-manifolds

  • Claude LeBrunEmail author
Article

Abstract

If \(M\) is the underlying smooth oriented four-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics \(h\) on \(M\) such that \(W^+(\omega , \omega )> 0\), where \(W^+\) is the self-dual Weyl curvature of \(h\), and \(\omega \) is a non-trivial self-dual harmonic two-form on \((M,h)\). While this open region in the space of Riemannian metrics contains all the known Einstein metrics on \(M\), we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on \(M\).

Keywords

Einstein metric Del Pezzo surface Weyl curvature Moduli space Harmonic form Kähler Almost-Kähler Symplectic 

Mathematics Subject Classification

53C25 (Primary) 14J26 32J15 53C55 53D05 

Notes

Acknowledgments

The author would like to thank Tedi Draghici for subsequently pointing out some of his own related work, and the anonymous referee for suggesting ways to streamline and clarify the exposition. This work was supported in part by NSF Grant DMS-1205953.

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsState University of New YorkStony BrookUSA

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