By Werner Ballmann

Those notes are according to lectures the writer gave on the collage of Bonn and the Erwin Schrödinger Institute in Vienna. the purpose is to provide an intensive advent to the idea of Kähler manifolds with distinct emphasis at the differential geometric facet of Kähler geometry. The exposition starts off with a brief dialogue of advanced manifolds and holomorphic vector bundles and a close account of the elemental differential geometric houses of Kähler manifolds. The extra complex themes are the cohomology of Kähler manifolds, Calabi conjecture, Gromov's Kähler 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 ecu Mathematical Society (EMS). allotted in the Americas via the yankee Mathematical Society.

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**Extra resources for Lectures on Kähler Manifolds (Esi Lectures in Mathematics and Physics)**

**Example text**

36, (∗ ⊗ h)α is of type (m − p, m − q). Therefore ((∗ ⊗ h)α) ∧ε β is a complex valued differential form of type (m, m − 1). It follows that ∂ ((∗ ⊗ h)α) ∧ε β = d ((∗ ⊗ h)α) ∧ε β . Since the real dimension of M is 2m, we have (∗ ⊗ h∗ )(∗ ⊗ h) = (−1)r on forms of degree r = p + q. 46), we get d ((∗ ⊗ h)α) ∧ε β = ∂ ((∗ ⊗ h)α) ∧ε β = (−1)2m−r {−((∗ ⊗ h∗ )∂(∗ ⊗ h)α, β) + (α, ∂β)} vol. It follows that ∂ ∗ = (∗ ⊗ h∗ )∂(∗ ⊗ h). 9) Here the ∂-operator on the right belongs to the dual bundle E ∗ of E. Let be the Laplace operator associated to ∂.

As for the second, we have (∇X J)Y, Z = ∇X (JY ), Z − J(∇X Y ), Z = ∇X (JY ), Z + ∇X Y, JZ . By the Koszul formula and the definition of ω, 2 ∇X (JY ), Z = X JY, Z + JY X, Z − Z X, JY = Xω(Y, Z) − JY ω(JZ, X) + Zω(X, Y ) and 2 ∇X Y, JZ = X Y, JZ + Y X, JZ − JZ X, Y = −Xω(JY, JZ) + Y ω(Z, X) − JZω(X, JY ), where we use that X, Y, Z, JY , and JZ commute. 17 Theorem. 33) and Levi-Civita connection ∇. Then the following assertions are equivalent: 48 ¨ hler Manifolds Lectures on Ka 1. g is a K¨ ahler metric.

Then the complex structure Jp turns Tp M into a complex vector space. Since J is parallel, the holonomy group Hol M = Holp M of M at p preserves Jp .