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Neighborhoods of Analytic Varieties: César Camacho, Hossein Movasati
In the first chapter we will review some will-known facts and definitions. The notion of reduced analytic variety, embedding dimension of singularities, formal neighborhood and obstruction to the existence of a formal isomorphism between two embeddings are discussed in this chapter. Cartan's theorem on the quotient of analytic varieties and Remmert reduction Theorem are presented. One of the main theorems in this chapter is Theorem 1.5. This theorem establishes the obstructions to the extension of a finite isomorphism of neighborhoods to a higher order isomorphism. The second chapter is devoted to pseudoconvex domains. For some technical reasons, we have preferred to work with C2 convex functions instead of C2 plurisubharmonic functions. A convex function carries just the convexity information of its level varieties and is easy to handle, so we use convex functions rather than plurisubharmonic functions. Theorem 2.2 reveals an important cohomological property of pseudoconvex domains. It can be considered as Cartan's B theorem for Stein varieties. Using Remmert reduction theorem on pseudoconvex domains, one can see that pseudoconvex domains are the point modification of Stein varieties. This leads to the notion of exceptional or negatively embedded varieties.
One of the natural examples of an embedded manifold is the zero section of a vector bundle. We deal with these embedding in chapter three. The zero section of a line bundle is an exceptional variety if and only the line bundle is negative in the sense of Kodaira, Theorem 3.1. Vanishing theorems for the germ of exceptional varieties are stated in this Chapter, Theorems 3.3, 3.5.
Chapter four is devoted to the formal principle. Theorem 1.5 and Theorem 3.3 give us a formal isomorphism of two negatively embedded manifolds. Roughly speaking, the formal principle tells us when a formal isomorphism of two neighborhoods implies the existence of a biholomorphism. In this chapter we have stated Artin's Theorem 4.1. This theorem implies the formal principle for singularities, Theorem 4.2, and then the formal principle for exceptional varieties can be derived.
Chapter five is devoted to foliated neighborhoods. In the first steps we will consider the most simple foliations which and transversal foliations. The main theorem in this direction is Theorem 5.1. Next, foliations with tangencies and Poincaré type singularities is considered. We generalize Grauert's step by step extension of isomorphisms to the case where the germ of embedding is foliated. In this section wealso introduce the notion of formal equivalence of two foliated neighborhoods and prove Theorem 5.3. Artin in [Art68] after stating his extension and lifting theorems poses the following question: Can one generalize these statements in various ways by requiring the map preserve extra structure, such as a stratification? We are interested in this case where this additional structure is a foliation.
Whenever it was possible, we have used figures to help to understand a definition, a theorem or its proof. Specially we hope that the figures will help on reconstructing the proofs in the mind. At the end of each chapter we have added some lines for the reader who wants to know more on the development of the material presented in the chapter. This will be useful also for classrooms activities.