Stein Manifolds

Cartan in 1953 formulated Theorems A and B, while Oka from 1936 to 1953 proved them for domains in C^n, completing the revolution in multidimensional complex analysis. These theorems became the foundation of derived algebraic geometry.

• Derived algebraic geometry: infinity-categories of coherent sheaves on Stein manifolds
• D-modules: Hormander dbar estimates give regularity for PDE systems
• Quantum mechanics: Bergman spaces on Stein manifolds as quantization models

Предварительные знания

• Previous lesson

Definition and Examples of Stein Manifolds

Stein in 1951 axiomatized the class of complex manifolds where multidimensional analysis works as in C^1: holomorphic functions separate points and locally define coordinates. Oka-Cartan (Cartan theorems A and B, 1953) completely describes sheaf cohomology on such manifolds.

Hormander's Theorem and the dbar Problem

Hormander (1965) proved: on a strictly pseudoconvex domain D with weight phi, the equation dbar u = f is solvable with L^2 estimate ||u||^2 <= int |f|^2 e^{-phi} / lambda_min, where lambda_min is the smallest eigenvalue of the Levi form. This is the key tool in several complex variables and the basis for proving Cartan's theorems.

Key Ideas

• Stein manifold: holomorphically convex + holomorphically separable
• Theorem A: coherent sheaf is generated by global sections
• Theorem B: H^q(X,F) = 0 for q >= 1, F coherent
• Embedding: X^n embeds in C^{ceil(3n/2)+1}
• Hormander estimates: ||u||^2 <= int |f|^2 / lambda_min on strictly pseudoconvex domains

Further Directions

These ideas open paths to deeper mathematics.

• ca-29-complex-alg-geom — extends

Вопросы для размышления

• Give a concrete example.
• How does this connect to other areas of mathematics?