Hermitian spin surfaces with small eigenvalues of the Dolbeault operator

Bogdan Alexandrov

Displaying similar documents to “Hermitian spin surfaces with small eigenvalues of the Dolbeault operator”

The Dolbeault operator on Hermitian spin surfaces

Bodgan Alexandrov, Gueo Grantcharov, Stefan Ivanov (2001)

Annales de l’institut Fourier

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We prove the vanishing of the kernel of the Dolbeault operator of the square root of the canonical line bundle of a compact Hermitian spin surface with positive scalar curvature. We give lower estimates of the eigenvalues of this operator when the conformal scalar curvature is non -negative.

Locally conformally Kähler metrics on Hopf surfaces

Paul Gauduchon, Liviu Ornea (1998)

Annales de l'institut Fourier

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A primary Hopf surface is a compact complex surface with universal cover $ℂ^{2} - {(0, 0)}$ and cyclic fundamental group generated by the transformation $(u, v) \mapsto (α u + λ v^{m}, β v)$ , $m \in ℤ$ , and $α, β, λ \in ℂ$ such that $∣ α ∣ \geq ∣ β ∣ > 1$ and $(α - β^{m}) λ = 0$ . Being diffeomorphic with $S^{3} \times S^{1}$ Hopf surfaces cannot admit any Kähler metric. However, it was known that for $λ = 0$ and $∣ α ∣ = ∣ β ∣$ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for primary Hopf surfaces of class $1$ ( $λ = 0$ )....

Weakly-Einstein hermitian surfaces

Vestislav Apostolov, Oleg Muškarov (1999)

Annales de l'institut Fourier

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A consequence of the Riemannian Goldberg-Sachs theorem is the fact that the Kähler form of an Einstein Hermitian surface is an eigenform of the curvature operator. Referring to this property as we obtain a complete classification of the compact locally homogeneous $*$ -Einstein Hermitian surfaces. We also provide large families of non-homogeneous $*$ -Einstein (but non-Einstein) Hermitian metrics on $ℂ ℙ^{2} ♯ {\overline{ℂ ℙ}}^{2}$ , $ℂ ℙ^{1} \times ℂ ℙ^{1}$ , and on the product surface $X \times Y$ of two curves $X$ and $Y$ whose genuses are greater than 1...

Self-duality of Kähler surfaces

Mitsuhiro Itoh (1984)

Compositio Mathematica

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Scalar-flat Kähler metrics on blown-up ruled surfaces.

Claude LeBrun (1991)

Journal für die reine und angewandte Mathematik

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Construction of Kähler surfaces with constant scalar curvature

Yann Rollin, Michael Singer (2009)

Journal of the European Mathematical Society

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Scalar-flat Kähler surfaces of all genera.

Jongsu Kim, C., Pontecorvo, M. LeBrun (1997)

Journal für die reine und angewandte Mathematik

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Toric Hermitian surfaces and almost Kähler structures

Włodzimierz Jelonek (2007)

Annales Polonici Mathematici

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The aim of this paper is to investigate the class of compact Hermitian surfaces (M,g,J) admitting an action of the 2-torus T² by holomorphic isometries. We prove that if b₁(M) is even and (M,g,J) is locally conformally Kähler and χ(M) ≠ 0 then there exists an open and dense subset U ⊂ M such that $(U, g_{| U})$ is conformally equivalent to a 4-manifold which is almost Kähler in both orientations. We also prove that the class of Calabi Ricci flat Kähler metrics related with the real Monge-Ampère equation...

Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections

Julien Keller, Christina Tønnesen-Friedman (2012)

Open Mathematics

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We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric.