Introduction to Several Complex Variables and Complex Manifolds

Navigating the transition from one complex variable to several requires a deep understanding of multidimensional complex space and geometric structures. This text-based course offers a clear, structured path through the mathematical foundations of complex manifolds and their analysis. You will build a strong intuitive grasp of how holomorphic functions behave in higher dimensions and how these local properties shape global geometry. By studying core theorems and modern algebraic tools, you will learn to analyze complex structures with confidence. What you'll learn: - Understand the core differences between one and several complex variables, starting with foundational definitions and domains of holomorphy - Explore the structure of complex manifolds and the transition functions that define them - Apply harmonic theory and the dbar-operator to solve differential equations on complex domains - Analyze the Hodge decomposition theorem and its implications for cohomology groups - Study the Hard Lefschetz theorem and how it relates the topology of Kähler manifolds to their algebraic structure - Evaluate vanishing theorems to determine the existence of holomorphic sections on vector bundles This course begins with a thorough introduction to essential terminology, complex coordinate systems, and basic sheaf theory. You will then progress through the standard machinery of differential forms, Kähler geometry, and the deep topological theorems that govern complex manifolds. This course is designed for advanced undergraduate or beginning graduate students in mathematics who have a solid background in real analysis, basic complex analysis, and elementary topology, but no prior exposure to multidimensional complex geometry is required. Prepare to elevate your mathematical reasoning and expand your geometric toolkit today.