# Hodge theory and complex algebraic geometry 1 by Voisin C. By Voisin C.

Similar algebraic geometry books

Introduction to modern number theory : fundamental problems, ideas and theories

This variation has been known as ‘startlingly up-to-date’, and during this corrected moment printing you'll be convinced that it’s much more contemporaneous. It surveys from a unified standpoint either the trendy country and the tendencies of continuous improvement in a number of branches of quantity conception. Illuminated via effortless difficulties, the critical principles of contemporary theories are laid naked.

Singularity Theory I

From the stories of the 1st printing of this ebook, released as quantity 6 of the Encyclopaedia of Mathematical Sciences: ". .. My common impact is of a very great e-book, with a well-balanced bibliography, steered! "Medelingen van Het Wiskundig Genootschap, 1995". .. The authors provide right here an up to the moment consultant to the subject and its major purposes, together with a couple of new effects.

An introduction to ergodic theory

This article offers an advent to ergodic thought compatible for readers realizing simple degree concept. The mathematical necessities are summarized in bankruptcy zero. it's was hoping the reader might be able to take on examine papers after examining the publication. the 1st a part of the textual content is worried with measure-preserving changes of likelihood areas; recurrence homes, blending homes, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy concept are mentioned.

Additional info for Hodge theory and complex algebraic geometry 1

Example text

48 2 Complex Manifolds Clearly φ is then a submersion, and each ﬁbre φ −1 (v) is a closed submanifold of U having the property that its tangent space at each point is equal to the ﬁbre of E at that point. The following theorem characterises the integrable distributions. 20 (Frobenius) A distribution E is integrable if and only if for all C 1 vector ﬁelds χ , ψ contained in E, the bracket [χ, ψ] is also contained in E. Proof Obviously, if E is integrable, then E is stable under the bracket. e. 19.

Another essential application is the principle of analytic continuation. 22 Let U be a connected open set of Cn , and f a holomorphic function on U . If f vanishes on an open set of U , then f is identically zero. e. locally equal to the sum of its Taylor series). We can thus apply the principle of analytic continuation to f . We recall that the latter is shown by noting that if f is analytic, the open set consisting of the points in whose neighbourhood f vanishes is equal to the closed set consisting of the points where f and all its derivatives vanish.

Let 1 ∈ R, 0 < 1 < |z 1 | be such that the closed disk of radius 1 and centre z 1 is contained in the disk {ζ | |ζ | < r1 }. Then the polydisk D 1 := {(ζ1 , . . , ζn )| |ζ1 − z 1| ≤ 1, |z i | ≤ ri , i ≥ 2} is contained in D − {ζ1 = 0}, so that Cauchy’s formula gives f (z) = 1 2iπ n f (ζ ) ∂D 1 dζ1 dζn ∧ ··· ∧ , ζ1 − z 1 ζn − z n where ∂ D 1 := {(ζ1 , . . , ζn )| |ζ1 − z 1| = 1, |ζi | = ri , i ≥ 2}. 2 Holomorphic functions of several variables 33 Consider, also, the product of circles ∂ D := {(ζ1 , .