By T. Ochiai

ISBN-10: 0120010186

ISBN-13: 9780120010189

Complicated stories in natural arithmetic, quantity 18-I: contemporary issues in Differential and Analytic Geometry offers the advancements within the box of analytical and differential geometry. This publication presents a few generalities approximately bounded symmetric domains.

Organized into components encompassing 12 chapters, this quantity starts off with an outline of harmonic mappings and holomorphic foliations. this article then discusses the worldwide constructions of a compact Kähler manifold that's in the community decomposable as an isometric made from Ricci-positive, Ricci-negative, and Ricci-flat components. different chapters ponder the main well-known non-standard examples of compact homogeneous Einstein manifolds developed through Riemannian submersions. This ebook discusses besides the usual compactification of the moduli house of polarized Einstein–Kähler orbitfold with a given Hilbert polynomials. the ultimate bankruptcy bargains with fixing a degenerate Monge–Ampère equation through developing a relatives of Einstein–Kähler metrics at the delicate a part of minimum different types of common kind.

This publication is a worthwhile source for graduate scholars and natural mathematicians.

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**Extra info for Recent Topics in Differential and Analytic Geometry**

**Sample text**

Let X be an η-dimensional compact complex manifold such that for some positive 1 q holomorphic line bundle L we have Κ χ = L for some integer q > n + 1 . n Then X is biholomorphic to the projective space P . 2) makes use of the Riemann-Roch Theorem and the Kodaira Vanishing Theorem. Let now ( X , h) be a compact Kahler manifold of positive holomorphic bisectional curvature. By the Bochner formula of Bishop-Goldberg (cf. 2)) we know that 6 2 ( X ) = 1. From the Universal Coefficient 2 Theorem it follows that # ( X , Z) = Z.

P . When dr > dim f c(Aut(5)) + l r we obtain non-trivial deformations of Dr = ($p) (D). To find rational curves one considers deformations of D'r which fixes some point χ G D'r corresponding to Ρ G 5. The deformations are then described by the subsheaf TT of the normal bundle ND>r\x> defined by local holomorphic sections vanishing at x. Since deg(^" r) = deg(7V£M|x') — η for r large enough it is always possible to obtain such a non-trivial deformation. If in so doing all the curves remained irreducible one would then have a projective-algebraic curve Γ and a mor- Topics in Complex Differential Geometry 45 phism F:S χ Γ - » X such that F(S χ Γ ) is a surface and such that F({P} χ Γ ) = {χ}.

Let M be a compact Kahler manifold; Ν be n an irreducible compact quotient of A ,n > 2. Suppose f:M —• Ν is a harmonic map. , Fn). , a submersion) at some point and that f is of rank at least 3 at some point. Then, each Fi is either holomorphic or anti-holomorphic. Remark. 2) was already stated in Mok [Mok6]. There we proved the existence of a closed leaf L of Τ χ in Μ — Ζ, but as indicated above there was a gap in deducing that Fi is either holomorphic or anti-holomorphic. Let L be the closure of L in M.

### Recent Topics in Differential and Analytic Geometry by T. Ochiai

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