9.3 Oka's coherence theorem

Oka's coherence theorem is a key result in complex analysis and sheaf theory. It establishes when a sheaf of holomorphic functions is coherent, which is crucial for understanding the structure of holomorphic functions on complex manifolds.

The theorem requires the complex manifold to be Stein, a type that behaves well with holomorphic functions. This condition allows local properties of holomorphic functions to be extended globally, enabling powerful techniques from homological algebra in complex analysis.

Oka's coherence theorem

• Fundamental result in complex analysis and sheaf theory
• Establishes conditions under which a sheaf of holomorphic functions is coherent
• Plays a crucial role in understanding the structure of holomorphic functions on complex manifolds

Definition of coherence

Top images from around the web for Definition of coherence

• 

  Hallmarking quantum states: unified framework for coherence and correlations – Quantum View original

  Is this image relevant?

• 

  analysis:course-w16:week11 [] View original

  Is this image relevant?

• 

  Compositional resource theories of coherence – Quantum View original

  Is this image relevant?

• 

  Hallmarking quantum states: unified framework for coherence and correlations – Quantum View original

  Is this image relevant?

• 

  analysis:course-w16:week11 [] View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition of coherence

• 

  Hallmarking quantum states: unified framework for coherence and correlations – Quantum View original

  Is this image relevant?

• 

  analysis:course-w16:week11 [] View original

  Is this image relevant?

• 

  Compositional resource theories of coherence – Quantum View original

  Is this image relevant?

• 

  Hallmarking quantum states: unified framework for coherence and correlations – Quantum View original

  Is this image relevant?

• 

  analysis:course-w16:week11 [] View original

  Is this image relevant?

1 of 3

• Property of a sheaf $\mathcal{F}$ on a complex manifold $X$
• Requires that for every open set $U \subset X$ and every finite collection of sections $s_1, \ldots, s_n \in \mathcal{F}(U)$, there exists an exact sequence: $\mathcal{O}_X^p|_U \to \mathcal{O}_X^q|_U \to \mathcal{F}|_U \to 0$ where $\mathcal{O}_X$ denotes the sheaf of holomorphic functions on $X$
• Intuitively, coherence means that local relations among sections can be extended globally

Conditions for coherence

• Oka's theorem provides sufficient conditions for a sheaf to be coherent
• Main condition: the complex manifold $X$ must be Stein
  • Stein manifolds are complex manifolds that behave well with respect to holomorphic functions
  • Examples include $\mathbb{C}^n$, complex submanifolds of $\mathbb{C}^n$, and domains of holomorphy
• Additional technical conditions on the sheaf $\mathcal{F}$ (e.g., locally finitely generated)

Importance in complex analysis

• Coherence is a key property in the study of holomorphic functions on complex manifolds
• Allows for the extension of local properties of holomorphic functions to global properties
• Enables the use of powerful techniques from homological algebra and sheaf theory in complex analysis

Proof of Oka's theorem

• Involves several intricate steps and relies on deep results in complex analysis and sheaf theory
• Utilizes the properties of Stein manifolds and the Oka-Cartan principle

Key steps in proof

• Reduction to the case of a polydisk using local coordinates and the Oka-Cartan principle
• Construction of a suitable resolution of the sheaf $\mathcal{F}$ using the Dolbeault complex
• Establishing the exactness of the resolution using the $\overline{\partial}$-Poincaré lemma and the properties of Stein manifolds
• Concluding the coherence of $\mathcal{F}$ from the exactness of the resolution

Techniques used

• Oka-Cartan principle: allows for the extension of local holomorphic functions to global ones on Stein manifolds
• Dolbeault complex: a resolution of the sheaf of holomorphic functions using differential forms
• $\overline{\partial}$-Poincaré lemma: a key result in the theory of the Dolbeault complex
• Homological algebra: used to study the exactness of sequences of sheaves

Applications of Oka's theorem

• Oka's coherence theorem has numerous applications in complex analysis and related fields
• Enables the solution of various problems involving holomorphic functions on complex manifolds

Extending holomorphic functions

• Oka's theorem allows for the extension of holomorphic functions defined locally to global holomorphic functions
• Provides conditions under which a holomorphic function defined on a subset of a complex manifold can be extended to the entire manifold
• Useful in the study of domains of holomorphy and the Levi problem

Solving Cousin problems

• Cousin problems involve finding holomorphic functions with prescribed zeros and poles on a complex manifold
• Oka's theorem can be used to solve certain types of Cousin problems
• Provides a framework for understanding the obstruction to solving Cousin problems in terms of cohomology groups

Relation to other theorems

• Oka's coherence theorem is closely related to several other important results in complex analysis and sheaf theory
• These theorems often work in tandem to provide a comprehensive understanding of holomorphic functions on complex manifolds

Cartan's theorems A and B

• Cartan's theorem A states that the sheaf cohomology groups $H^q(X, \mathcal{F})$ vanish for $q > 0$ when $X$ is Stein and $\mathcal{F}$ is coherent
• Cartan's theorem B asserts that every holomorphic function on a closed complex submanifold of a Stein manifold can be extended to the entire manifold
• These theorems, along with Oka's coherence theorem, form the foundation of the theory of coherent analytic sheaves

Grauert's direct image theorem

• Grauert's theorem concerns the direct image sheaf $f_*\mathcal{F}$ of a coherent sheaf $\mathcal{F}$ under a proper holomorphic map $f$
• States that if $f: X \to Y$ is a proper holomorphic map between complex manifolds and $\mathcal{F}$ is a coherent sheaf on $X$, then $f_*\mathcal{F}$ is a coherent sheaf on $Y$
• Provides a powerful tool for studying the behavior of coherent sheaves under holomorphic mappings

Historical context

• Oka's coherence theorem was a groundbreaking result in the development of complex analysis and sheaf theory
• Proved by Japanese mathematician Kiyoshi Oka in the 1950s

Kiyoshi Oka's contributions

• Kiyoshi Oka (1901-1978) was a pioneering figure in the field of complex analysis
• Made significant contributions to the theory of several complex variables and the study of complex analytic spaces
• Oka's lemma and Oka's principle are named after him, highlighting his influence on the field

Influence on sheaf theory development

• Oka's coherence theorem played a crucial role in the development of sheaf theory, particularly in the context of complex analysis
• Provided a bridge between complex analysis and algebraic geometry, allowing for the application of algebraic techniques to analytic problems
• Inspired further research into coherent sheaves, leading to important results such as Serre's GAGA theorem and the Cartan-Serre-Grothendieck theorem