Special Kähler-Ricci potentials on compact Kähler manifolds
Identifieur interne : 005386 ( Main/Exploration ); précédent : 005385; suivant : 005387Special Kähler-Ricci potentials on compact Kähler manifolds
Auteurs : A. Derdzinski [États-Unis] ; G. Maschler [États-Unis]Source :
- Journal fur die reine und angewandte Mathematik (Crelles Journal) [ 0075-4102 ] ; 2006-04-25.
English descriptors
- Teeft :
- Analogous objects, Biholomorphic, Biholomorphic isometry, Biholomorphism, Boundary conditions, Bres, Bundle projection, Codimension, Compact kahler manifold, Compact kahler manifolds, Compact manifold, Complex codimension, Complex dimension, Complex dimensions, Complex line bundle, Complex manifold, Complex submanifold, Complex vector bundle, Complex vector space, Conformally, Conformally einstein, Connection form, Constant value, Crit, Critical manifold, Critical manifolds, Critical point, Critical points, Curvature form, Curvature tensor, Derdzinski, Eigenvalue, Eld, Endpoint, Euclidean, Expy, Extremal kahler metrics, Ftmin tmax, Geodesic, Geodesic segment, Geodesic segments, Hermitian, Hessy, Holomorphic, Holomorphic line bundle, Horizontal distribution, Inversion biholomorphism, Isometry, Kahler, Kahler form, Kahler manifold, Kahler manifolds, Kahler metrics, Leibniz rule, Lemma, Lemma derdzinski, Lemma lemma, Local model, Local models, Local result, Main result, Manifold, Maschler, Metric, Metrics, Nf0g, Nonconstant, Nontrivial, Nonzero, Nonzero eigenvalue, Norm function, Normal bundle, Normal connection, Normal space, Open subset, Positive function, Present paper, Product bundle, Real number, Ricci, Ricci tensor, Riemannian, Riemannian manifold, Second part, Single point, Special case, Special potentials, Stronger assumption, Structure theorem, Submanifold, Subset, Tangent bundle, Tensor, Tmax, Tmin, Tmin tmax, Total space, Unit sphere, Variable point, Vector bundle, Vertical distribution.
Abstract
By a special Kähler-Ricci potential on a Kähler manifold we mean a nonconstant real-valued C ∞ function τ such that J(∇τ) is a Killing vector field and, at every point with dτ ≠ 0, all nonzero tangent vectors orthogonal to ∇τ and J(∇τ) are eigenvectors of both ∇ dτ and the Ricci tensor. For instance, this is always the case if τ is a nonconstant C∞ function on a Kähler manifold (M, g) of complex dimension m > 2 and the metric g˜ = g/τ2, defined wherever τ ≠ 0, is Einstein. (When such τ exists, (M, g) may be called almost-everywhere conformally Einstein.) We provide a complete classification of compact Kähler manifolds (M, g) with special Kähler-Ricci potentials, showing, in particular, that in any complex dimension m ≧ 2 they form two separate classes: in one, M is the total space of a holomorphic ℂP1 bundle; in the other, M is biholomorphic to ℂP m . We then use this classification to prove a structure theorem for compact Kähler manifolds of any complex dimension m > 2 which are almost-everywhere conformally Einstein.
Url:
DOI: 10.1515/CRELLE.2006.030
Affiliations:
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<term>Biholomorphic isometry</term>
<term>Biholomorphism</term>
<term>Boundary conditions</term>
<term>Bres</term>
<term>Bundle projection</term>
<term>Codimension</term>
<term>Compact kahler manifold</term>
<term>Compact kahler manifolds</term>
<term>Compact manifold</term>
<term>Complex codimension</term>
<term>Complex dimension</term>
<term>Complex dimensions</term>
<term>Complex line bundle</term>
<term>Complex manifold</term>
<term>Complex submanifold</term>
<term>Complex vector bundle</term>
<term>Complex vector space</term>
<term>Conformally</term>
<term>Conformally einstein</term>
<term>Connection form</term>
<term>Constant value</term>
<term>Crit</term>
<term>Critical manifold</term>
<term>Critical manifolds</term>
<term>Critical point</term>
<term>Critical points</term>
<term>Curvature form</term>
<term>Curvature tensor</term>
<term>Derdzinski</term>
<term>Eigenvalue</term>
<term>Eld</term>
<term>Endpoint</term>
<term>Euclidean</term>
<term>Expy</term>
<term>Extremal kahler metrics</term>
<term>Ftmin tmax</term>
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<term>Lemma</term>
<term>Lemma derdzinski</term>
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<term>Local models</term>
<term>Local result</term>
<term>Main result</term>
<term>Manifold</term>
<term>Maschler</term>
<term>Metric</term>
<term>Metrics</term>
<term>Nf0g</term>
<term>Nonconstant</term>
<term>Nontrivial</term>
<term>Nonzero</term>
<term>Nonzero eigenvalue</term>
<term>Norm function</term>
<term>Normal bundle</term>
<term>Normal connection</term>
<term>Normal space</term>
<term>Open subset</term>
<term>Positive function</term>
<term>Present paper</term>
<term>Product bundle</term>
<term>Real number</term>
<term>Ricci</term>
<term>Ricci tensor</term>
<term>Riemannian</term>
<term>Riemannian manifold</term>
<term>Second part</term>
<term>Single point</term>
<term>Special case</term>
<term>Special potentials</term>
<term>Stronger assumption</term>
<term>Structure theorem</term>
<term>Submanifold</term>
<term>Subset</term>
<term>Tangent bundle</term>
<term>Tensor</term>
<term>Tmax</term>
<term>Tmin</term>
<term>Tmin tmax</term>
<term>Total space</term>
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<front><div type="abstract" xml:lang="en">By a special Kähler-Ricci potential on a Kähler manifold we mean a nonconstant real-valued C ∞ function τ such that J(∇τ) is a Killing vector field and, at every point with dτ ≠ 0, all nonzero tangent vectors orthogonal to ∇τ and J(∇τ) are eigenvectors of both ∇ dτ and the Ricci tensor. For instance, this is always the case if τ is a nonconstant C∞ function on a Kähler manifold (M, g) of complex dimension m > 2 and the metric g˜ = g/τ2, defined wherever τ ≠ 0, is Einstein. (When such τ exists, (M, g) may be called almost-everywhere conformally Einstein.) We provide a complete classification of compact Kähler manifolds (M, g) with special Kähler-Ricci potentials, showing, in particular, that in any complex dimension m ≧ 2 they form two separate classes: in one, M is the total space of a holomorphic ℂP1 bundle; in the other, M is biholomorphic to ℂP m . We then use this classification to prove a structure theorem for compact Kähler manifolds of any complex dimension m > 2 which are almost-everywhere conformally Einstein.</div>
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