By Raymond O. Wells Jr. (auth.)

In constructing the instruments useful for the research of advanced manifolds, this finished, well-organized remedy offers in its establishing chapters an in depth survey of contemporary growth in 4 parts: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations. next chapters then strengthen such subject matters as Hermitian external algebra and the Hodge *-operator, harmonic concept on compact manifolds, differential operators on a Kahler manifold, the Hodge decomposition theorem on compact Kahler manifolds, the Hodge-Riemann bilinear kinfolk on Kahler manifolds, Griffiths's interval mapping, quadratic variations, and Kodaira's vanishing and embedding theorems.

The 3rd version of this usual reference encompasses a new appendix through Oscar Garcia-Prada which provides an summary of sure advancements within the box in the course of the many years because the booklet first appeared.

From studies of the 2d Edition:

"..the new version of Professor Wells' booklet is well timed and welcome...an very good advent for any mathematician who suspects that complicated manifold concepts might be suitable to his work."

- Nigel Hitchin, Bulletin of the London Mathematical Society

"Its objective is to give the fundamentals of research and geometry on compact advanced manifolds, and is already one of many commonplace resources for this material."

- Daniel M. Burns, Jr., Mathematical Reviews

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**Extra info for Differential Analysis on Complex Manifolds**

**Example text**

A ⊗ B, the tensor product. Hom(A, B), the linear maps from A to B. A∗ , the linear maps from A to K. 5, we can extend all the above algebraic constructions to vector bundles. For example, suppose that we have two vector bundles πE : E −→ X Then deﬁne and πF : F −→ X. Ep ⊕ F p . E⊕F = p∈X We then have the natural projection π: E ⊕ F −→ X given by π −1 (p) = Ep ⊕ Fp . Now for any p ∈ X we can ﬁnd a neighborhood U of p and local trivializations ∼ hE : E|U −→U × K n ∼ hF : F |U −→U × K m , and we deﬁne hE⊕F : E ⊕ F |U −→ U × (K n ⊕ K m ) by hE⊕F (v + w) = (p, hpE (v) + hpF (w)) for v ∈ Ep and w ∈ Fp .

5: Let E −→ X be an S-bundle of rank r and let U be an open subset of X. A frame for E over U is a set of r S-sections {s1 , . . , sr }, sj ∈ S(U, E), such that {s1 (x), . . , sr (x)} is a basis for Ex for any x ∈ U . Any S-bundle E admits a frame in some neighborhood of any given point in the base space. Namely, let U be a trivializing neighborhood for E so that ∼ h: E|U −→U × K r , and thus we have an isomorphism ∼ h∗ : S(U, E|U )−→S(U, U × K r ). Consider the vector-valued functions e1 = (1, 0, .

3) i d d 0 −→ R −→ E0X −→ E1X −→ · · · −→ Em X −→ 0, where i is the natural inclusion and d is the exterior differentiation operator. Since d 2 = 0, it is clear that the above is a differential sheaf. , Spivak [1], p. 94) asserts that on a star-shaped domain U in Rn , if f ∈ Ep (U ) is given such that df = 0, then there exists a u ∈ Ep − 1 (U ) (p > 0) so that du = f . Therefore the induced mapping dx on the stalks at x ∈ X is exact, since we can ﬁnd representatives in local coordinates in star-shaped domains.