Download Lectures on Kähler Geometry (London Mathematical Society by Andrei Moroianu PDF

By Andrei Moroianu

Kähler geometry is a gorgeous and interesting sector of arithmetic, of considerable study curiosity to either mathematicians and physicists. This self-contained 2007 graduate textual content offers a concise and obtainable advent to the subject. The publication starts off with a assessment of simple differential geometry, ahead of relocating directly to an outline of complicated manifolds and holomorphic vector bundles. Kähler manifolds are mentioned from the viewpoint of Riemannian geometry, and Hodge and Dolbeault theories are defined, including an easy facts of the well-known Kähler identities. the ultimate a part of the textual content reviews a number of elements of compact Kähler manifolds: the Calabi conjecture, Weitzenböck thoughts, Calabi-Yau manifolds, and divisors. All sections of the e-book finish with a sequence of workouts and scholars and researchers operating within the fields of algebraic and differential geometry and theoretical physics will locate that the ebook presents them with a valid realizing of this thought.

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Additional resources for Lectures on Kähler Geometry (London Mathematical Society Student Texts, Volume 69)

Example text

For every linear connection ∇, we will denote by the same symbol the covariant derivative induced by ∇ on the tensor bundle which satisfies the Leibniz rule with respect to the tensor product, commutes with contractions and equals the usual vector derivative on functions. 16, we need to show that there exists a unique linear connection ∇ such that T ∇ = 0 and ∇g = 0. The first relation reads ∇X Y − ∇Y X = [X, Y ], ∀ X, Y ∈ X (M ). 1) In order to exploit the second relation, we notice that if X, Y and Z are vector fields on M , we can write ∂X (g(Y, Z)) = ∇X (g(Y, Z)) = ∇X (C1 C2 (g ⊗ Y ⊗ Z)) = C1 C2 (∇X (g ⊗ Y ⊗ Z)) = (∇X g)(Y, Z) + g(∇X Y, Z) + g(Y, ∇X Z).

A connection H on P induces canonically a pull-back connection f ∗ H on f ∗ P by the formula (f ∗ H)(u,x) := {(V, X) ∈ Hu × Tx M | π∗ (V ) = f∗ (X)}. 8, we define f ∗ ∇ to be the covariant derivative on f ∗ E which induces the connection f ∗ H on f ∗ Gl(E) = Gl(f ∗ E). 5. Parallel transport Let π : P → M be a G-structure with a connection H on it. A path ut in P is called horizontal if u˙ t ∈ Hut for all t. 12. For every smooth path x : [0, 1] → M , t → xt and u ∈ P such that π(u) = x0 , there exists a unique horizontal path ut in P such that π(ut ) = xt for all t ∈ [0, 1].

We first prove a local version of the statement. On the pull-back bundle x∗ P (which is a G-structure over I) let X denote the horizontal lift of the standard vector field ∂/∂t ∈ X (I) with respect to the pull-back connection. 11, for every t0 ∈ I and u ∈ π −1 (xt0 ), there exists a unique integral curve (t, ut ) of X in x∗ P defined on some open interval Ut0 containing t0 . 3) shows that ut is horizontal in P . 6. Holonomy 43 ut a is also horizontal for every a ∈ G. Consequently, the open set of definition Ut0 of the local horizontal lifts defined above does not depend on the element u of the fibre π −1 (xt0 ).

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