Complex analytic and differential geometry by Demailly J.-P.

By Demailly J.-P.

Show description

Read Online or Download Complex analytic and differential geometry PDF

Best differential geometry books

Geometric Phases in Classical and Quantum Mechanics

This paintings examines the gorgeous and demanding actual thought often called the 'geometric phase,' bringing jointly various actual phenomena below a unified mathematical and actual scheme. numerous well-established geometric and topological tools underscore the mathematical therapy of the topic, emphasizing a coherent standpoint at a slightly subtle point.

Lectures on Symplectic Geometry

Discusses differential geometry and hyperbolic geometry. For researchers and graduate scholars. Softcover.

Differential Geometry and Topology: With a View to Dynamical Systems

Available, concise, and self-contained, this booklet deals an excellent advent to 3 comparable topics: differential geometry, differential topology, and dynamical platforms. themes of precise curiosity addressed within the booklet contain Brouwer's fastened element theorem, Morse conception, and the geodesic circulate.

Additional resources for Complex analytic and differential geometry

Sample text

Then, for every point y ∈ ∂B, there exists a holomorphic function f ∈ Ç(X) such that f (y) = f (x0). Replacing f with λ(f − f (x0)), we can achieve f (x0) = 0 and |f (y)| > 1. By compactness of ∂B, we find finitely many functions f1 , . . , fN ∈ Ç(X) such that v0 = |fj |2 satisfies v0 (x0 ) = 0, while v0 1 on ∂B. Now, we set u0 (z) = v0 (z) on X 2 Mε {v0 (z), (|z| + 1)/3} on B. 18. Then u0 is smooth and plurisubharmonic, coincides with v0 near ∂B and with (|z|2 + 1)/3 on a neighborhood of x0 .

Let (Ωα ) be a locally finite open covering of Ω by relatively compact open balls contained in coordinate patches of X. Choose concentric balls Ω′′α ⊂ Ω′α ⊂ Ωα of respective radii rα′′ < rα′ < rα and center z = 0 in the given coordinates z = (z1 , . . , zn ) near Ωα , such that Ω′′α still cover Ω. We set uα (z) = u ⋆ ρεα (z) + δα (rα′2 − |z|2 ) on Ωα . For εα < εα,0 and δα < δα,0 small enough, we have uα u + λ/2 and Huα on Ωα . Set ηα = δα min{rα′2 − rα′′2 , (rα2 − rα′2 )/2}. (1 − λ)γ Choose first δα < δα,0 such that ηα < minΩα λ/2, and then εα < εα,0 so small that ′′ u u ⋆ ρεα < u + ηα on Ωα .

18) Lemma. For arbitrary η = (η1 , . . , ηp ) ∈ ]0, +∞[p, the function Mη (t1 , . . , tp ) = Rn max{t1 + h1 , . . , tp + hp } θ(hj /ηj ) dh1 . . dhp 1 j n possesses the following properties: a) Mη (t1 , . . , tp ) is non decreasing in all variables, smooth and convex on Rn ; b) max{t1 , . . , tp } Mη (t1 , . . , tp ) max{t1 + η1 , . . , tp + ηp } ; c) Mη (t1 , . . ,ηp ) (t1 , . . , tj , , . . , tp ) if tj + ηj maxk=j {tk − ηk } ; d) Mη (t1 + a, . . , tp + a) = Mη (t1 , . . , tp ) + a, ∀a ∈ R ; e) if u1 , .

Download PDF sample

Rated 4.28 of 5 – based on 13 votes