## IMPA - Complex Analysis: Multiple Variables

This is a course for students who mastered complex analysis in one variable, basic topology and theory of smooth manifolds, and want to know about complex analysis of multiple variables. The main subject of these lectures is the local parametrization of complex varieties analogue of Noether lemma. Sheaves, manifolds, complex manifolds, holomorphic functions, Cauchy formula in one and many variables.

Limits and colimits. Germs of continuous, smooth and holomorphic functions. Weierstrass preparation theorem. Weierstrass divisibility theorem. Noetherian rings. A number of monographs of various aspects of complex analysis in several variables have appeared since the first version of this book was published, but none of them uses the analytic techniques based on the solution of the Neumann Problem as the main tool.

The additions made in this third, revised edition place additional stress on results where these methods are particula A number of monographs of various aspects of complex analysis in several variables have appeared since the first version of this book was published, but none of them uses the analytic techniques based on the solution of the Neumann Problem as the main tool.

The additions made in this third, revised edition place additional stress on results where these methods are particularly important.

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The local arguments in this section are closely related to the proof of the coherence of the sheaf of germs of functions vanishing on an analytic set. Also added is a discussion of the theorem of Siu on the Lelong numbers of plurisubharmonic functions. Since the L2 techniques are essential in the proof and plurisubharmonic functions play such an important role in this book, it seems natural to discuss their main singularities.

Get A Copy. Hardcover , pages. Published January 16th by North-Holland first published January 1st More Details Original Title. Other Editions 4. Friend Reviews. To see what your friends thought of this book, please sign up. Lists with This Book. This book is not yet featured on Listopia. Community Reviews. Showing There are both canonical reproducing kernels and constructible reproducing kernels.

These objects are quite different, but there are important connections between the two theories.

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## An Introduction to Complex Analysis in Several Variables

We touch on those connections. A big part of what makes the Bergman theory important is the invariance of the kernel and the metric. Here we explore this circle of ideas. As an application, we give a new proof of the biholomorphic inequivalence of the ball and the polydisc. This chapter serves as an entree to these ideas.

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One of the key features that makes analysis of several variables different from analysis of one variable is the role of geometry. In some cases this is nontrivial Riemannian geometry, and an entirely new point of view must be developed. In this chapter we develop this theme by way of the Berezin transform, the invariant Laplacian, and Bergman representative coordinates.

Ideas of Fefferman come into play in a decisive way. In this chapter we explore a variety of topics arising from the Bergman theory and also from the function theory of several complex variables. We construct and analyze the worm domain, and comment on the Bergman theory on the worm. We look at situations in which the Bergman kernel and projection become pathological. We study the boundary behavior of the Bergman kernel.

We present a famous example of David Barrett, and we explore the use of plurisubharmonic functions. Many of the tools and constructs of one complex variable are not available in the several complex variable setting. As a substitute in several complex variables we have sheaf theory and partial differential equations.

In the present chapter we develop the latter topic.