By Werner Ballmann

Those notes are in accordance with 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 speculation of Kähler manifolds with precise 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 an in depth account of the fundamental 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 international research and partial differential equations is believed, particularly 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). dispensed in the Americas through the yankee Mathematical Society.

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

**Example text**

M ∈ H(M ) such that ϕ0 (q) = 0 for all q ∈ U and such that the holomorphic map z : U → Cm given by ϕj = z j ϕ0 defines a holomorphic coordinate chart of M on U . We may have M0 = ∅. In holomorphic coordinates z on an open subset U of M , write 2 θ = im f dz 1 ∧ · · · ∧ dz m ∧ dz 1 ∧ · · · ∧ dz m . Then f is a non-negative real analytic function on U and f > 0 on U ∩ M0 . The differential form ω := i∂∂ ln f on U ∩ M0 does not depend on the choice of coordinates, hence ω is a real analytic differential form of type (1, 1) on M0 .

V2k )| ≤ k! volV (v1 , . . , v2k ) with equality iff V is a complex linear subspace of W . Proof. Observe first that both sides of the asserted inequality are multiplied by | det A| if we transform (v1 , . . , v2k ) with A ∈ Gl(2k, R) to another basis of V . Hence we are free to choose a convenient basis of V . Let v, w ∈ W be orthonormal unit vectors with respect to ·, · . Then ω(v, w) = Jv, w ∈ [−1, 1] and ω(v, w) = ±1 iff w = ±Jv. This shows the assertion in the case k = 1. In the general case we observe first that the restriction of ω to V is given by a skew-symmetric endomorphism A of V .

For a section σ of E write σ = σ ′ + σ ′′ , where σ ′ is a section of E ′ and σ ′′ is perpendicular to E ′ . Let D be the Chern connection on E with respect to h. Then the Chern connection D′ of E ′ with respect to h′ is given by D′ σ = (Dσ)′ . Recall that this is the standard recipe of getting a Hermitian connection for a subbundle of a Hermitian bundle with a Hermitian connection. 16, applied to the trivial bundle E = M × Ck . 21 Proposition. Let E → M be a holomorphic vector bundle with Hermitian metric h and corresponding Chern connection D.