For every integer m > 1, the manifold #_m(S^2 x S^2) admits a T^2-invariant Einstein metric with positive Ricci curvature that is not Hermitian.
K\"ahler-Einstein metrics of negative curvature
8 Pith papers cite this work. Polarity classification is still indexing.
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abstract
Given any integer $n\geq 2$, we construct a compact K\"ahler-Einstein manifold of dimension n of negative sectional curvature which is not covered by the ball.
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For every k≥2, the manifolds #k(S2×S2) and kCP2#(2k+1)CPbar2 admit smooth positive Einstein metrics, so infinitely many homeomorphism types do.
For compact Einstein four-manifolds, the spectral gap α>β in the self-dual Weyl curvature forces β=γ=−α/2, so the metric is conformally Kähler with positive scalar curvature (after at most a double cover).
Constructs complete Kähler metrics with negative bisectional curvature on hyperbolic complex manifolds resolving Mok's problem and projective surfaces with negative HSC realizing any rational Chern slope in (2/7, 2/3).
Cyclic branched covers of complex hyperbolic manifolds have non-complex-hyperbolic Chern number ratios, proven exactly in dimension 2 and claimed with a gap in higher even dimensions.
A model Einstein metric on complex hyperbolic branched covers, built the way Fine and Premoselli built theirs in the real hyperbolic case, is shown to equal the model Kähler-Einstein metric of Guenancia and Hamenstädt and to produce negatively curved Einstein metrics in every complex dimension.
Expository overview of the theory of Einstein 4-manifolds, covering the Hitchin-Thorpe inequality, Seiberg-Witten obstructions, monopole classes, and constructions of Einstein metrics on various smooth 4-manifolds.
Branched covers of closed hyperbolic manifolds along totally geodesic codimension-2 submanifolds possess specific geometric properties that are compiled from previously known results.
citing papers explorer
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T^2-invariant Einstein metrics on closed 4-manifolds
For every integer m > 1, the manifold #_m(S^2 x S^2) admits a T^2-invariant Einstein metric with positive Ricci curvature that is not Hermitian.
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Positive Einstein metrics on infinitely many four-manifolds
For every k≥2, the manifolds #k(S2×S2) and kCP2#(2k+1)CPbar2 admit smooth positive Einstein metrics, so infinitely many homeomorphism types do.
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Conformal K\"{a}hler rigidity of Einstein four-manifolds
For compact Einstein four-manifolds, the spectral gap α>β in the self-dual Weyl curvature forces β=γ=−α/2, so the metric is conformally Kähler with positive scalar curvature (after at most a double cover).
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Curvature of hyperbolic complex manifolds
Constructs complete Kähler metrics with negative bisectional curvature on hyperbolic complex manifolds resolving Mok's problem and projective surfaces with negative HSC realizing any rational Chern slope in (2/7, 2/3).
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On ratios of Chern numbers for complex hyperbolic branched covers
Cyclic branched covers of complex hyperbolic manifolds have non-complex-hyperbolic Chern number ratios, proven exactly in dimension 2 and claimed with a gap in higher even dimensions.
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An explicit description of the K\"{a}hler-Einstein metrics of Guenancia-Hamenst\"{a}dt
A model Einstein metric on complex hyperbolic branched covers, built the way Fine and Premoselli built theirs in the real hyperbolic case, is shown to equal the model Kähler-Einstein metric of Guenancia and Hamenstädt and to produce negatively curved Einstein metrics in every complex dimension.
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An Overview of Einstein 4-Manifolds
Expository overview of the theory of Einstein 4-manifolds, covering the Hitchin-Thorpe inequality, Seiberg-Witten obstructions, monopole classes, and constructions of Einstein metrics on various smooth 4-manifolds.
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The geometry of branched coverings of hyperbolic manifolds
Branched covers of closed hyperbolic manifolds along totally geodesic codimension-2 submanifolds possess specific geometric properties that are compiled from previously known results.