Several complex variables and CR geometry

For a proper holomorphic map F from a ball B n into another ball B N , we give a criterion when a rational map F is equivalent to a polynomial one.As application, we show that proper rational holomorphic maps from B 2 into B N of degree two are equivalent to polynomial maps, and we also give an example of a rational holomorphic maps of degree 3 that are 'almost' linear but are not equivalent to holomorphic polynomial maps.

We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.

In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transformation formula for the reduced Bergman kernels under proper holomorphic maps. We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

Applications holomorphes propres d'un domaine quasi-disqué sur un domaine disqué

A complex variable is a quantity that contains both a real part and an imaginary part. Complex analysis is an application of mathematics that analyzes the features of functions of complex variables. Physics, engineering, and computer science are just a few of the scientific disciplines that benefit from complex analysis. In order to tackle issues that are challenging or impossible to resolve using only real variables, complex analysis is crucial. This paper introduces an important theorem in complex analysis, which is Cauchy’s residue theorem. A relative general expression for a complex integral along a simple closed contour is provided by Cauchy's residue theorem. Cauchy's residue theorem can be used to select an acceptable closed contour for the calculation of some unusual definite integrals that may be highly complex and challenging to solve using traditional methods. The residue at the function's isolated singularities is then established. In the formula that is subtracted in Cauchy's residue theorem, the values of the residues are extracted. The integral along a simple closed contour can then be represented in two pieces, in which one along the real axis and the other along the circle.

A number of results concerning infinite series are presented in this chapter. The material spans convergence tests for series of constants, uniform convergence for series of functions, and power series. It is not the intention here to give a comprehensive account of this theory, but rather to review material that is relevant to later chapters. This chapter is intended to make the rest of the book more self contained and remind the reader of some basic results. It is assumed that the reader has a background in real analysis of a single variable and elementary complex analysis; consequently, no attempt is made to prove these results or provide examples. Proofs and examples can be found in textbooks on real analysis such as Apostol (Mathematical Analysis, Addison-Wesley, 2nd ed., 1974), Fulks (Advanced Calculus, 3rd ed., Wiley, 1978), and Little et al. (Real Analysis via Sequences and Series, Springer-Verlag, 2015) and complex analysis (Ahlfors, Complex Analysis, McGraw-Hill, 2nd ed., 1966; Brown and Churchill, Complex Variables and Applications, McGraw-Hill, 7th ed., 2004; Priestley, Introduction to Complex Analysis, Rev. ed., Oxford University Press, 1990). The exception is in the final section on double sequences and series. This material is often not part of courses on real or complex analysis. Here we provide proofs, examples and exercises.

We construct a functorial pushforward homomorphism in geometric Hodge filtered complex cobordism along proper holomorphic maps between arbitrary complex manifolds. This significantly improves previous results on such transfer maps and is a much stronger result than the ones known for differential cobordism of smooth manifolds. This enables us to define and provide a concrete geometric description of Hodge filtered fundamental classes for all proper holomorphic maps. Moreover, we give a geometric description of a cobordism analog of the Abel–Jacobi invariant for nullbordant maps which is mapped to the classical invariant under the Hodge filtered Thom morphism. For the latter we provide a new construction in terms of geometric cycles.

We show that if E is a closed convex set in {mathbb {C}}^n(n>1) contained in a closed halfspace H such that Ecap bH is nonempty and bounded, then the concave domain Omega = {mathbb {C}}^n{setminus } E contains images of proper holomorphic maps f:Xrightarrow {mathbb {C}}^n from any Stein manifold X of dimension <n, with approximation of a given map on closed compact subsets of X. If in addition 2dim X+1le n then f can be chosen an embedding, and if 2dim X=n, then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties.

This study is to determine the location of the difficulty of complex variable courses. Complex variable courses require a strong understanding, high concentration, good learning readiness, reasoning power and higher order logic in understanding direct objects. The direct objects referred to are Facts, Concepts, Skills, and Principles. In addition to direct objects in obtaining maximum results, indirect objects are needed. Indirect objects that are meant are tools used in the form of media, support and encouragement from others in obtaining good results. The results of the mid-semester grades for the 2014/2015 academic year were about 13% of students who got good grades and the rest were not good. In fact, this complex variable course is the main course of several courses. Another fact is that in the 2019/2020 semester VII, there are 35% in the poor category and 50% sufficient and the rest are good and very good. From 35% of students with poor learning outcomes took remedial, 1 person graduated with sufficient marks, while the rest did not pass. The specific objectives of the research are 1) to determine the mastery in complex variable courses, 2) the location of the difficulty and difficulty factors of complex variables, 3) to know the mastery and difficulty of indirect objects. This type of research approach is descriptive qualitative. Research wants to describe and describe how the actual event. The research was conducted in the Complex Variable Analysis course, Semester VII, implementation October-January and a total of 12 subjects. Techniques for obtaining data from observations, interviews and documentation. The analysis technique, presenting data, reducing and drawing conclusions. Conclusion,The difficulty of students lies in the facts and concepts that exist in the direct object. In the interview session, 71.42% said that the difficulty factor was the mastery of facts and concepts. In online implementation, students experience percentage difficulties when writing symbols and operating the media used. This is a new problem for mathematics students in achieving good learning outcomes with objective assessment standards.

The local residue has been presented since the early days of several complex variables and plays a fundamental role in complex analysis and geometry. Nevertheless, in many cases, there are very few practical tools in the computational point of view.

Let Gi be closed Lie groups of U(n), Ωi be bounded Gi-invariant domains in ℂn which contains 0, and O(ℂn)Gi = ℂ, for i = 1, 2. It is known that if f: Ω1 →} Ω2 is a proper holomorphic mapping, and f−1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f.

This chapter develops basic properties of complex Euclidean space. Some of the main ideas are unitary transformations, the holomorphic automorphism group of the unit ball, the use of Hermitian forms, and proper holomorphic mappings. We also gather some elementary combinatorial information.

Denote by \(\triangle\) the open unit disc in \({\Bbb C}\). We prove that given a discrete subset S of a connected Stein manifold M there is a proper holomorphic map \(f:\triangle\to M\) such that \(S\subset f(\triangle)\); if \(\dim M\ge 3\) the map f can be chosen to be an embedding. In addition we prove that we can prescribe higher order contacts of \(f(\triangle)\) with given one dimensional submanifolds in M.

We prove that any proper holomorphic mapping between two equidimensional irreducible bounded symmetric domains with rank ≥ 2 \geq 2 is a biholomorphism. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics.

Proper holomorphic maps between balls of different dimensions.