Kodaira dimension and the Yamabe problem

@article{Lebrun1997KodairaDA,
  title={Kodaira dimension and the Yamabe problem},
  author={Claude Lebrun},
  journal={Communications in Analysis and Geometry},
  year={1997},
  volume={7},
  pages={133-156}
}
  • C. Lebrun
  • Published 13 February 1997
  • Mathematics
  • Communications in Analysis and Geometry
The Yamabe invariant Y (M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar curvature Riemannian metrics g on M . (To be absolutely precise, one only considers constant-scalar-curvature metrics which are Yamabe minimizers, but this does not affect the sign of the answer.) If M is the underlying smooth 4-manifold of a complex algebraic surface (M,J), it is shown that the sign of Y (M) is completely determined by the Kodaira dimension… Expand
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References

SHOWING 1-10 OF 30 REFERENCES
Scalar curvature of a metric with unit volume
The problem of finding Riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of Kazdan and Warner in 1970's ([10] andExpand
The smooth invariance of the Kodaira dimension of a complex surface
Introduction The purpose of this note is to announce the following result: Theorem 1. Let X be a complex surface of general type. Then X is not diffeomorphic to a rational surface. Combining TheoremExpand
Polarized 4-Manifolds, Extremal Kahler Metrics, and Seiberg-Witten Theory
Using Seiberg-Witten theory, it is shown that any Kahler metric of constant negative scalar curvature on a compact 4-manifold M minimizes the L 2 -norm of scalar curvature among Riemannian metricsExpand
On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*
Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent theExpand
Conformal deformation of a Riemannian metric to constant scalar curvature
A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. ThisExpand
On the scalar curvature of complex surfaces
Let (M,J) be a minimal compact complex surface of Kaehler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a KAEHLER metric ofExpand
Yamabe Constants and the Perturbed Seiberg-Witten Equations
Among all conformal classes of Riemannian metrics on ${\Bbb CP}_2$, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of theExpand
THE SEIBERG-WITTEN INVARIANTS AND SYMPLECTIC FORMS
(Note: There are no symplectic forms on X unless b and the first Betti number of X have opposite parity.) In a subsequent article with joint authors, a vanishing theorem will be proved for theExpand
Four-Manifolds without Einstein Metrics
It is shown that there are infinitely many compact simply con- nected smooth 4-manifolds which do not admit Einstein metrics, but nev- ertheless satisfy the strict Hitchin-Thorpe inequality 2 !> 3|"Expand
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3
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