By Demailly J.-P.

**Read or Download Complex analytic and differential geometry PDF**

**Similar differential geometry books**

**Connections, curvature and cohomology. Vol. III: Cohomology by Werner Hildbert Greub PDF**

Greub W. , Halperin S. , James S Van Stone. Connections, Curvature and Cohomology (AP Pr, 1975)(ISBN 0123027039)(O)(617s)

Ranging from undergraduate point, this booklet systematically develops the fundamentals of - research on manifolds, Lie teams and G-manifolds (including equivariant dynamics) - Symplectic algebra and geometry, Hamiltonian platforms, symmetries and aid, - Integrable platforms, Hamilton-Jacobi idea (including Morse households, the Maslov category and caustics).

**A treatise on the geometry of surfaces by Alfred Barnard Basset PDF**

This quantity is made out of electronic photos from the Cornell collage Library old arithmetic Monographs assortment.

**Download e-book for iPad: Riemannian Geometry by Peter Petersen (auth.)**

Meant for a 12 months path, this article serves as a unmarried resource, introducing readers to the $64000 concepts and theorems, whereas additionally containing adequate heritage on complicated subject matters to entice these scholars wishing to specialise in Riemannian geometry. this can be one of many few Works to mix either the geometric components of Riemannian geometry and the analytic features of the idea.

**Extra resources for Complex analytic and differential geometry**

**Sample text**

Definition and Basic Properties of Subharmonic Functions 32 Chapter I. 1. Definition. Mean Value Inequalities. If u is a Borel function on B(a, r) which is bounded above or below, we consider the mean values of u over the ball or sphere: 1 u(x) dλ(x), αm r m B(a,r) 1 u(x) dσ(x). 9) r 1 αm r m σm−1 tm−1 µS (u ; a, t) dt 0 1 tm−1 µS (u ; a, rt) dt. 5) with x = 0. We get Pr (0, y) = 1/σm−1 r m−1 and Gr (0, y) = r (|y|2−m − r 2−m )/(2 − m)σm−1 = −(1/σm−1 ) |y| t1−m dt, thus B(0,r) ∆u(y) Gr (0, y) dλ(y) = − r 1 σm−1 1 =− m 0 dt tm−1 r ∆u(y) dλ(y) |y|

B) If (Ωα )α∈I is a family of pseudoconvex open subsets of Cn , the interior of the inter◦ section Ω = is pseudoconvex. α∈I Ωα c) If (Ωj )j∈N is a non decreasing sequence of pseudoconvex open subsets of Cn , then Ω = j∈N Ωj is pseudoconvex. Proof. a) Let ϕ, ψ be smooth plurisubharmonic exhaustions of Ω, Ω′ . Then (z, w) −→ ϕ(z) + ψ(w) is an exhaustion of Ω × Ω′ and z −→ ϕ(z) + ψ(F (z)) is an exhaustion of F −1 (Ω′ ). b) We have − log d(z, ∁Ω) = supα∈I − log d(z, ∁Ωα ), so this function is plurisubharmonic.

If u ∈ C 2 (Ω, R), the subharmonicity of restrictions of u to complex lines, C ∋ w −→ u(a + wξ), a ∈ Ω, ξ ∈ Cn , is equivalent to ∂2 u(a + wξ) = ∂w∂w 1 j,k n ∂ 2u (a + wξ) ξj ξ k ∂zj ∂z k 0. Therefore, u is plurisubharmonic on Ω if and only if ∂ 2 u/∂zj ∂z k (a) ξj ξ k is a semipositive hermitian form at every point a ∈ Ω. 8) Theorem. If u ∈ Psh(Ω), u ≡ −∞ on every connected component of Ω, then for all ξ ∈ Cn ∂ 2u Hu(ξ) := ξj ξ k ∈ ′ (Ω) ∂zj ∂z k 1 j,k n is a positive measure. Conversely, if v ∈ ′ (Ω) is such that Hv(ξ) is a positive measure for every ξ ∈ Cn , there exists a unique function u ∈ Psh(Ω) locally integrable on Ω such that v is the distribution associated to u.

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

by Ronald

4.2