# alpha brooks Book Archive

Geometry And Topology

# Complex Analytic and Differential Geometry (September 2009 by Jean-Pierre Demailly By Jean-Pierre Demailly

Best geometry and topology books

Zeta Functions, Topology and Quantum Physics

This quantity specializes in numerous facets of zeta capabilities: a number of zeta values, Ohno’s family, the Riemann speculation, L-functions, polylogarithms, and their interaction with different disciplines. 11 articles on contemporary advances are written by means of notable specialists within the above-mentioned fields. each one article begins with an introductory survey resulting in the fascinating new study advancements finished through the participants.

Extra info for Complex Analytic and Differential Geometry (September 2009 draft)

Example text

Geometric Characterizations of Pseudoconvex Open Sets We first discuss some characterizations of pseudoconvex open sets in Cn . We will need the following elementary criterion for plurisubharmonicity. 1) Criterion. Let v : Ω −→ [−∞, +∞[ be an upper semicontinuous function. Then v is plurisubharmonic if and only if for every closed disk ∆ = z0 + D(1)η ⊂ Ω and every polynomial P ∈ C[t] such that v(z0 + tη) Re P (t) for |t| = 1, then v(z0 ) Re P (0). Proof. The condition is necessary because t −→ v(z0 + tη) − Re P (t) is subharmonic in a neighborhood of D(1), so it satisfies the maximum principle on D(1) by Th.

All properties are immediate consequences of the definition, except perhaps e). That Mη (u1 , . . , up ) is plurisubharmonic follows from a) and Th. 6. Fix a point z0 and ε > 0. All functions u′j (z) = uj (z) − γz0 (z − z0 ) + ε|z − z0 |2 are plurisubharmonic near z0 . It follows that Mη (u′1 , . . , u′p ) = u − γz0 (z − z0 ) + ε|z − z0 |2 is also plurisubharmonic near z0 . Since ε > 0 was arbitrary, e) follows. 19) Corollary. Let uα ∈ ∞ (Ωα ) ∩ Psh(Ωα ) where Ωα ⊂⊂ X is a locally finite open covering of X.

B. Kiselman’s Minimum Principle We already know that a maximum of plurisubharmonic functions is plurisubharmonic. However, if v is a plurisubharmonic function on X × Cn , the partial minimum function on X defined by u(ζ) = inf z∈Ω v(ζ, z) need not be plurisubharmonic. A simple counterexample in C × C is given by v(ζ, z) = |z|2 + 2 Re(zζ) = |z + ζ|2 − |ζ|2 , u(ζ) = −|ζ|2 . It follows that the image F (Ω) of a pseudoconvex open set Ω by a holomorphic map F need not be pseudoconvex. In fact, if Ω = {(t, ζ, z) ∈ C3 ; log |t| + v(ζ, z) < 0} and if Ω′ ⊂ C2 is the image of Ω by the projection map (t, ζ, z) −→ (t, ζ), then Ω′ = {(t, ζ) ∈ C2 ; log |t| + u(ζ) < 0} is not pseudoconvex.