By T. Ochiai
Complex experiences in natural arithmetic, quantity 18-I: contemporary subject matters in Differential and Analytic Geometry offers the advancements within the box of analytical and differential geometry. This ebook offers a few generalities approximately bounded symmetric domains.
Organized into components encompassing 12 chapters, this quantity starts with an outline of harmonic mappings and holomorphic foliations. this article then discusses the worldwide buildings 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 contemplate the main well-known non-standard examples of compact homogeneous Einstein manifolds developed through Riemannian submersions. This booklet discusses in addition the average compactification of the moduli house of polarized Einstein–Kähler orbitfold with a given Hilbert polynomials. the ultimate bankruptcy offers with fixing a degenerate Monge–Ampère equation by means of developing a relatives of Einstein–Kähler metrics at the tender a part of minimum forms of normal kind.
This publication is a useful source for graduate scholars and natural mathematicians.
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Example text
We construct moduli spaces by introducing an equivalence relation 9ft; such that Ti C 9ft; C defined as the (non-empty) union of complex-analytic components of We show that 9ft; is indeed an equivalence relation and open by making use of the fact that Ti is a foliation of complex-analytic hypersurfaces. Given Kaup's result the heuristic argument can indeed be made rigorous to rule out Case (i). Case (ii) This is the situation when some leaf of T\ is not closed. The key idea of Mok [MoklO] is to exploit a recurrent point of the foliation to generate a one-parameter group G of automorphisms of the unit disc Δ ι so that when one moves on a connected component of a level set S\ of the function λ on Μ - Ζ (obtained by lifting the function λ on Topics in Complex Differential Geometry 33 M — Z\ the image under F\ lies inside an orbit of the one-parameter group G.
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. To complete the proof given in [Mok6], the key point is to show that η _1 / * ( T T I ( L ) ) C Hx ( Α ^ ( Δ ) ) for some finite subgroup of A u t ( Δ ι ) .
Denote the image of πχ(Μ) on Aut(Ex) by Ξ. It follows by using the invariance property of the Poincaré metric that there exists a sequence { & } in Ξ converging to the identity. From this one deduces that every point on Σ χ is a point of accumulation of Ξ. Equivalently, every point of Μ — Ζ is a recurrent point of T\. Fix now a point x G M - Ζ at which Fx is a submersion. Write χ for the projection of χ into Μ — Ζ. Let 5 be a the local piece of the level set S\ passing through x. S traces a smooth local curve C on Δχ passing through Ρχ(χ), which we may take to be the origin o.
