By Gabor Szekelyhidi
A uncomplicated challenge in differential geometry is to discover canonical metrics on manifolds. the simplest identified instance of this is often the classical uniformization theorem for Riemann surfaces. Extremal metrics have been brought via Calabi as an try out at discovering a higher-dimensional generalization of this consequence, within the surroundings of Kahler geometry. This ebook provides an creation to the examine of extremal Kahler metrics and specifically to the conjectural photograph touching on the lifestyles of extremal metrics on projective manifolds to the soundness of the underlying manifold within the feel of algebraic geometry. The ebook addresses many of the simple principles on either the analytic and the algebraic facets of this photo. an summary is given of a lot of the required historical past fabric, akin to uncomplicated Kahler geometry, second maps, and geometric invariant concept. past the elemental definitions and homes of extremal metrics, numerous highlights of the idea are mentioned at a degree available to graduate scholars: Yau's theorem at the life of Kahler-Einstein metrics, the Bergman kernel enlargement as a result of Tian, Donaldson's decrease certain for the Calabi power, and Arezzo-Pacard's life theorem for consistent scalar curvature Kahler metrics on blow-ups.
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Additional resources for An Introduction to Extremal Kahler Metrics
Proof. 5) from our earlier calculation now implies (in our changed notation) that 3. Kahler-Einstein Metrics 46 for some constants£, C1 > 0, since we are assuming that g and g are uniformly equivalent. Using the previous lemma, we can then choose a large constant A such that ~(ISl 2 + Atr9g) ~ ISl 2 - C2, for some C2. Suppose now that ISl 2 +Atr9g achieves its maximum at p EM. Then 0 ~ IBl 2 (p) - C2, 2 so ISl (p) ~ C2. Then at every other point x EM we have ISl 2(x) ~ ISl 2(x) + Atr9g(x) ~ ISl 2 (p) + Atr9g(p) ~ C2 +Ca, for some Ca, which is what we wanted to prove.
In particular we have functions Ak (analogous to the earlier Christoffel symbols) defined by \lks = Aks. Then the curvature is determined by (remembering that s is holomorphic) so Fkf = -qAk. To determine Ak we use the defining properties of the Chern connection to get 8kh(s) = (\lks, s)h = Akh(s), so Ak = h(s)- 18kh(s). It follows that Fkf = -q(h(s)- 18kh(s)) = -8z8k log h(s). We can summarize these calculations as follows. 1. 34. The curvature of the Chem connection of a holomorphic line bundle equipped with a Hermitian metric is given by Fkz = -8kqlog h(s), where h(s) = (s, s)h for a local holomorphic sections.
In addition we can assume that g' is diagonal at p since any Hermitian matrix can be diagonalized by a unitary transformation. -. t-,-. t g''PP g'JJ.... p,j = -B(tr91g)(tr9g'), where Bis the largest of the numbers -8p8pgjj (more geometrically-Bis a lower bound for the bisectional curvature of g). t Jpa g p,j,a L g''PP(8p9~a)(8pg~b). p,a,b 3. J . a,3,p since in the last sum we are simply adding in some non-negative terms. Finally, using the Kahler condition Op9jo. 6). 8. 4) satisfies c- 1 (9jk:) < (gjk + 8j8k:cp) < C(gjk:)· Proof.