Gabor Szekelyhidi's An Introduction to Extremal Kahler Metrics PDF

By Gabor Szekelyhidi

ISBN-10: 1470410478

ISBN-13: 9781470410476

A simple challenge in differential geometry is to discover canonical metrics on manifolds. the easiest identified instance of this can be the classical uniformization theorem for Riemann surfaces. Extremal metrics have been brought by way of Calabi as an try out at discovering a higher-dimensional generalization of this consequence, within the atmosphere of Kahler geometry. This ebook offers an creation to the research of extremal Kahler metrics and specifically to the conjectural photo pertaining to the lifestyles of extremal metrics on projective manifolds to the steadiness of the underlying manifold within the feel of algebraic geometry. The publication addresses a number of the simple principles on either the analytic and the algebraic aspects of this photo. an outline is given of a lot of the mandatory historical past fabric, reminiscent of uncomplicated Kahler geometry, second maps, and geometric invariant idea. past the elemental definitions and houses of extremal metrics, numerous highlights of the idea are mentioned at a degree obtainable to graduate scholars: Yau's theorem at the life of Kahler-Einstein metrics, the Bergman kernel enlargement because of Tian, Donaldson's reduce sure for the Calabi strength, and Arezzo-Pacard's lifestyles theorem for consistent scalar curvature Kahler metrics on blow-ups.

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Additional info for An Introduction to Extremal Kahler Metrics

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9. 3. 8, the c 0-norm off is controlled by the L 1-norm off together with the C°'-norm of Lf. This can be done by using a blow-up argument similar to what we have used above (a good exercise for the reader), although the more standard way is to use similar estimates in Sobolev spaces, together with the Sobolev embedding theorem. 3) for harmonic functions. An important point which does not follow from these arguments is that we do not need to know a priori that f E Ck+ 2•°'. 3) holds. For this, one needs to work harder; see Gilbarg-Trudinger [58, Chapter 6).

Generalize the previous exercise to higher-order derivatives of the curvature, IVkRml. 2) without appealing to the Schauder estimates. 4. The case c1 (M) = 0 When the manifold M has vanishing first Chern class, then a Kahler-Einstein metric on Mis necessarily Ricci flat. Given any metric won M, the Ricci form of w is exact, so by the 88-lemma there is a function F such that Ric(w) = Ra8F. a8cp to be Ricci flat, we need to solve the equation (w + Aaacpr = eF wn. A slight difference from earlier is that for this to be possible, we first need to normalize F by adding a constant.

31), so using this theorem, it is possible to construct many examples of Einstein manifolds. at. Finally we briefly discuss the case c1(M) > 0, which has only recently been solved. The algebrogeometric obstructions that appear in this and the more general case of extremal metrics will be our subject of study in the remainder of the book. The basic reference for this chapter is Yau [122), but there are many places where this material is explained, for instance Siu [97), Tian [113), or Blocki [17].

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An Introduction to Extremal Kahler Metrics by Gabor Szekelyhidi

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