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# Positivity in complex spaces and plurisubharmonic functions = Postivité dans les espaces complexes et fonctions plurisousharmoniques by Pierre Lelong

Written in English

## Subjects:

• Functions of complex variables.,
• Plurisubharmonic functions.,
• Differential forms.,
• Analytic functions.

Edition Notes

## Book details

Classifications The Physical Object Other titles Positivité dans les espaces complexes et fonctions plurisousharmoniques Statement by Pierre Lelong. Series Queen"s papers in pure and applied mathematics ;, v. 112, Queen"s papers in pure and applied mathematics ;, no. 112. LC Classifications QA3 .Q38 no. 112 Pagination x, 243 p. ; Number of Pages 243 Open Library OL116404M ISBN 10 0889118280 LC Control Number 99458593 OCLC/WorldCa 41867650

Add tags for "Positivity in complex spaces and plurisubharmonic functions = Postivité dans les espaces complexes et fonctions plurisousharmoniques". Be the first. Similar Items. You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read.

Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. not, and the plurisubharmonic functions form a class which is biholomorphically invariant and can be de ned on any complex analytic manifold.

A basic example of a plurisubharmonic function is clog jhj, where c is a positive constant and h is holomorphic. Thus the plurisubharmonic functions generalize the (absolute values of) holomorphic functions.

Subsequent sections examine plurisubharmonic functions and pseudoconvex domains in Banach spaces, along with Riemann domains and envelopes of holomorphy.

In addition to its value as a text for advanced graduate students of mathematics, this volume also functions as a reference for researchers and by: In Section existence theorems for the Cauchy-Riemann equations in several complex variables are proved in such sets for L 2 spaces with respect to weights e − ϕ where ϕ is plurisubharmonic.

This gives the tools required in Section to study the Lelong numbers of plurisubharmonic functions, describing the dominating associated mass.

POSITIVE CURRENTS AND PLURISUBHARMONIC FUNCTIONS 3 Proof. 1) Let us ﬁrst observe that if α is positive then it is Hermitian. Indeed, for any (n−1,n−1) simple positive form γ we have α ∧γ = α ∧γ = α ∧γ By Proposition the same is true for any (n−1,n−1) form.

Therefore α = ¯α. DOI: /_22 Corpus ID: Plurisubharmonic functions and potential theory in several complex variables @inproceedings{KiselmanPlurisubharmonicFA, title={Plurisubharmonic functions and potential theory in several complex variables}, author={C.

Kiselman}, year={} }. Given a complex manifold M, one says that plurisubharmonic, resp. holomorphic, domination is possible in M if for any locally bounded function u:M →R there is a continuous plurisub-harmonic function w: M → R, resp. a Banach space (V, V) and a holomorphic function h:M →V, such that u(x) w(x), resp.

u(x) h(x) V, for every x ∈M. Your proof of (ii) looks ok. You should perhaps mention that you're using the monotone convergence theorem at the end. For the corresponding statement about uniform convergence, you can use a very similar argument to establish the sub-mean value inequality as the one you use for (ii).

It touches all essential parts of complex function theory, and very often goes deeper into the subject than most elementary texts. a ] I find the pace, style and didactics in the presentation perfect. a ] All in all, this is one of the best if not the best book on the elementary theory of complex functions, and it can serve as a textbook as Reviews: Complex Diﬀerential Calculus and Pseudoconvexity This introductive chapter is mainly a review of the basic tools and concepts which will be employed in the rest of the book: diﬀerential forms, currents, holomorphic and plurisubharmonic functions, holo-morphic convexity and pseudoconvexity.

The Riesz functional associated to a plurisubharmonic function v on X takes f to 1 2 R D log j Delta j Delta(v ffi f), where Delta(v ffi f) is considered as a positive Borel measure on D, equal. have local potentials – plurisubharmonic functions.

So we start with the notion of Lelong number for plurisubharmonic functions. Plurisubharmonic functions Throughout the exposition, Ω is a domain in Cn, n>1, and uis a plurisubharmonic (psh) function in Ω, i.e., an upper semicontinuous function whose restriction to each complex.

To find some examples, try to construct a few functions whose complex Hessian is $$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \qquad \begin{pmatrix} 2 & 0 \\ 0 & -1 \end{pmatrix} \qquad \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \qquad$$ to get examples of a plurisubharmonic, subharmonic (but not psh) and non-subharmonic function.

in the cone of plurisubharmonic functions on G. Corollary. If moreover H2(G,R) = 0, the positive cone generated over rational coeﬃcients by currents of integration [D] on irreducible divisors of G is dense in the cone of closed positive currents of type (1,1) on G. The original proof of Lelong rests upon a use of complex function theory on.

Let φ ˜ = φ ∘ F, then φ ˜ is a plurisubharmonic function on Ω 1 \ S which have 0 as an upper bound. By Lemmaφ ˜ extends to a plurisubharmonic function on Ω 1 which attains its maximum on S. By the maximum principle of plurisubharmonic functions, φ ˜ is constant.

This is a contradiction. So S = ∅ and Ω 1 ′ = Ω 1. By comparing Green functions of multi-circled plurisubharmonic singularities in the n-domensional complex space to their indicators, we obtain formulas for the higher Lelong numbers and.

The notion of ω q-plurisubharmonic function has many facets, and these functions can be defined in many different main reason this notion is considered comes from the theory of plurisubharmonic functions on calibrated manifolds (see Subsection ).However, ω q-plurisubharmonic functions are very useful even outside of the theory of calibrations.

I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. Here are my questions, and I'd be grateful of any reference or information.

Let \$. Chapter 5 Legends. The heatmaps and simple annotations automatically generate legends which are put one the right side of the heatmap. By default there is no legend for complex annotations, but they can be constructed and added manually (Section ).All legends are internally constructed by Legend() constructor.

In later sections, we first introduce the settings for continuous legends and. [70] P. Lelong, Remarks on pointwise multiplicities, Linear Topologic Spaces and Complex Analysis 3 (), [71] P. Lelong, Positivity in complex spaces and plurisubharmonic functions.

Queen's Papers in Pure and Applied Mathematics, vol. Kingston/ Ontario,   Let φ be a plurisubharmonic function on Ω such that φ (t, α z) = φ (t, z) for α ∈ T n.

Then the function φ ⁎ defined as φ ⁎ (t): = inf z ∈ Ω t ⁡ φ (t, z) is a plurisubharmonic function on U. The argument in the proof of Theorem can be generalized to more general settings considered in. For example, in the same way one. Abstract: We study the question of when a -plurisubharmonic function on a complex manifold, where is a fixed -form, can be approximated by a decreasing sequence of smooth -plurisubharmonic functions.

We show in particular that it is always possible in the compact Kähler case. In this survey paper, we give a review of some recent developments on holomorphic invariants of bounded domains, which include squeezing functions, Fridman’s invariants, p-Bergman kernels, and spaces of $$L^p$$ integrable holomorphic functions.

plurisubharmonic Morse function φ all of whose critical points have index plurisubharmonic function is given by φ = φ0 + |z|2, where φ0 is an exhausting plurisubharmonic function on V and z is the complex.

We reduce the openness conjecture of Demailly and Kollár on the singularities of plurisubharmonic functions to a purely algebraic statement. Send article to Kindle To send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage.

Harmonic functions, for us, live on open subsets of real Euclidean spaces. Throughout this book, nwill denote a ﬁxed positive integer greater than 1 and Ω will denote an open, nonempty subset of Rn.A twice continuously diﬀerentiable, complex-valued function udeﬁned on Ω is harmonic on Ω if ∆u≡0, where∆ =D1 2++Dn 2 andDj.

We offer a tecnique to construct plurisubharmonic functions with prescribed order function and prove that for every G_delta-function f equal to a constant outside a countable set there is a maximal plurisubharmonic function whose order function coincides with f.

This shows that the space of fundamental solutions to the homogeneous Monge-Ampere. An illustration of an open book.

Books. An illustration of two cells of a film strip. Video. An illustration of an audio speaker. Audio. An illustration of a " floppy disk. Full text of "Analytic discs, plurisubharmonic hulls, and non-compactness of the d-bar-Neumann operator".

space of square integrable functions. In order of logical simplicity, the space. comes ﬁrst since it occurs already in the description of functions integrable in the Lebesgue sense. Connected to it via duality is the. L ∞ space of bounded functions, whose supremum norm carries over from the more familiar space of continuous functions.

CONTENTS Preface xiii Prologue: The Exponential Function Chapter 1 Abstract Integration 5 Set-theoretic notations and terminology 6 The concept of measurability 8 Simple functions 15 Elementary properties of measures 16 Arithmetic in [0, 00] 18 Integration of positive functions 19 Integration of complex functions 24 The role played by sets of measure zero complex-valued bounded functions on S.

Example Let Sbe a topological space (so that the notion of contin-uous functions from Sto R or to C makes sense) and let X = C(S), the collection of real-valued continuous functions on S. with addition and scalar multiplication being de ned by () and ().

As an application, it is shown that a connected noncompact complete Kähler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a 2-dimensional complex analytic set at some point has the Bochner–Hartogs property; that is, the first compactly supported.

Search the world's most comprehensive index of full-text books. My library. I hugely like this one, Complex Analysis (Princeton Lectures in Analysis, No. 2): Elias M. Stein, Rami Shakarchi: : Books Its not just an exceptionally good complex analysis book but it also provides a soft start towards.

2 Complex Functions and the Cauchy-Riemann Equations Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Like-wise, in complex analysis, we study functions f(z) of a complex variable z2C (or in some region of C).

Here we expect that f(z) will in general take values in C as well. sections) (problems in several complex variables and complex algebraic geometry) is reduced to the construction of speci ed plurisubharmonic functions (real-valued, may take 1) (problems.

Formal definition. A function: → ∪ {− ∞}, with domain ⊂ is called plurisubharmonic if it is upper semi-continuous, and for every complex line {+ ∣ ∈} ⊂ with, ∈the function ↦ (+) is a subharmonic function on the set {∈ ∣ + ∈}.In full generality, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space as follows.

Definition. In this article, the field of scalars denoted 𝔽 is either the field of real numbers ℝ or the field of complex numbers ℂ. Formally, an inner product space is a vector space V over the field 𝔽 together with a map ⋅, ⋅: × → called an inner product that satisfies the following conditions (1), (2), and (3) for all vectors x, y, z ∈ V and all scalars a ∈ 𝔽.

Exponential estimates for plurisubharmonic functions Dinh, Tien-Cuong, Nguyên, Viêt-Anh, and Sibony, Nessim, Journal of Differential Geometry, ; Extremal ω-plurisubharmonic functions as envelopes of disc functionals Magnússon, Benedikt Steinar, Arkiv för Matematik, ; On the Differentiability of Quaternion Functions Dzagnidze, Omar, Tbilisi Mathematical Journal.

COMPLEX NUMBERS Constructing the complex numbers One way of introducing the ﬁeld C of complex numbers is via the arithmetic of 2×2 matrices. DEFINITION A complex number is a matrix of the form x −y y x, where x and y are real numbers. Complex numbers of the form x 0 0 x are scalar matrices and are called.For each closed, positive (1,1)-current ω on a complex manifold X and each ω-upper semicontinuous function φ on X we associate a disc functional and prove that its envelope is equal to the supremum of all ω-plurisubharmonic functions dominated by is done by reducing to the case where ω has a global potential.

Then the result follows from Poletsky’s theorem, which is the special.2 SETS AND FUNCTIONS Subsets A set A is said to be a subset of a set B if every element of A is an element of B. We write A ⊂ B or B ⊃ A to indicate it and use expressions like A is contained in B.

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