By Werner Ballmann
Those notes are in line with lectures the writer gave on the collage of Bonn and the Erwin SchrÃ¶dinger Institute in Vienna. the purpose is to provide an intensive advent to the idea of KÃ¤hler manifolds with distinctive emphasis at the differential geometric facet of KÃ¤hler geometry. The exposition starts off with a quick dialogue of advanced manifolds and holomorphic vector bundles and an in depth account of the fundamental differential geometric homes of KÃ¤hler manifolds. The extra complicated themes are the cohomology of KÃ¤hler manifolds, Calabi conjecture, Gromov's KÃ¤hler hyperbolic areas, and the Kodaira embedding theorem. a few familiarity with worldwide research and partial differential equations is thought, specifically within the half at the Calabi conjecture. There are appendices on Chern-Weil concept, symmetric areas, and $L^2$-cohomology.
A ebook of the eu Mathematical Society (EMS). disbursed in the Americas by means of the yank Mathematical Society.
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Additional info for Lectures on Kaehler manifolds
G is a K¨ ahler metric. 2. dω = 0. 3. ∇J = 0. 4. In terms of holomorphic coordinates z, we have ∂gjk ∂z l = ∂glk ∂z j or, equivalently, ∂gjk ∂z l = ∂gjl ∂z k . 5. 3 is equal to the Levi-Civita connection ∇. 6. For each point p0 in M , there is a smooth real function f in a neighborhood of p0 such that ω = i∂∂f . 7. For each point p0 in M , there are holomorphic coordinates z centered at p0 such that g(z) = 1 + O(|z|2 ). A function f as in (6) will be called a K¨ ahler potential, holomorphic coordinates as in (7) will be called normal coordinates at p0 .
As for the third assertion, we have ∂θ = ∂(h−1 ∂h) = −(h−1 ∂hh−1 ) ∧µ ∂h = −(h−1 ∂h) ∧µ (h−1 ∂h) = −θ ∧µ θ. In particular, Θ = ∂θ + ∂θ + θ ∧µ θ = ∂θ. Hence ∂Θ = 0 and, by the Bianchi identity dΘ = Θ ∧λ θ, we conclude that Θ ∧λ θ = dΘ = ∂Θ + ∂Θ = ∂Θ. 23 Proposition. Let E → M be a holomorphic vector bundle with Hermitian metric h and Chern connection D. Let p0 ∈ M and z be holomorphic coordinates about p0 with z(p0 ) = 0. Then there is a holomorphic frame (Φ1 , . . , Φk ) of E about p0 such that 1) h(z) = 1 + O(|z|2 ); 2) Θ(0) = ∂∂h(0).
Xm , JXm ) of M , the Ricci tensor is given by Ric X = In particular, Ric JX = J Ric X. R(Xj , JXj )JX. 56 ¨ hler Manifolds Lectures on Ka Proof. We compute Ric(X, Y ) = R(Xj , X)Y, Xj + R(JXj , X)Y, JXj ) = R(Xj , X)JY, JXj − R(JXj , X)JY, Xj = R(X, Xj )JXj , JY + R(JXj , X)Xj , JY =− R(Xj , JXj )X, JY = R(Xj , JXj )JX, Y . We conclude that Ric(JX, JY ) = Ric(X, Y ). 58) and hence there is an associated real differential form of type (1, 1), the Ricci form ρ(X, Y ) := Ric(JX, Y ). 47). For the mixed terms, we have Ric(Zj , Zk ) = Rl ljk =− ∂Γl lk ∂zj = Ricjk = Ricjk = Ric(Zj , Zk ).