9.2 Analytic sheaves

Analytic sheaves are crucial tools in complex analytic geometry. They extend holomorphic functions to a global setting, allowing us to study complex manifolds and analytic spaces more flexibly. These sheaves encode local holomorphic data and enable the examination of global properties through section gluing.

Analytic sheaves share similarities with algebraic sheaves but differ in key aspects. Both types satisfy sheaf axioms and have similar structural properties. However, analytic sheaves use holomorphic functions, while algebraic sheaves use regular functions, leading to distinct behaviors in certain mathematical contexts.

Definition of analytic sheaves

• Analytic sheaves are a fundamental object of study in complex analytic geometry, extending the notion of holomorphic functions to a more global and flexible setting
• They provide a framework to study holomorphic functions and their properties on complex manifolds and analytic spaces
• Analytic sheaves encode local holomorphic data and allow for the study of global properties through the gluing of local sections

Holomorphic functions as sections

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• The sections of an analytic sheaf over an open set $U$ are the holomorphic functions defined on $U$
• The restriction maps between open sets are given by the restriction of holomorphic functions to smaller open sets
• The sheaf axioms ensure that local holomorphic functions can be glued together to obtain global sections, provided they agree on overlaps

Ringed space structure

• An analytic sheaf $\mathcal{O}_X$ on a complex manifold or analytic space $X$ endows $X$ with the structure of a ringed space
• The stalk $\mathcal{O}_{X,x}$ at each point $x \in X$ is a local ring, consisting of germs of holomorphic functions near $x$
• The ringed space structure allows for the study of local properties of analytic sheaves, such as coherence and finite generation

Morphisms between analytic sheaves

• Morphisms between analytic sheaves are defined as morphisms of sheaves that respect the holomorphic structure
• A morphism $\varphi: \mathcal{F} \to \mathcal{G}$ between analytic sheaves induces morphisms on the stalks $\varphi_x: \mathcal{F}_x \to \mathcal{G}_x$ for each $x \in X$
• Composition of morphisms and the identity morphism give the category of analytic sheaves on a complex manifold or analytic space

Analytic sheaves vs algebraic sheaves

• Analytic sheaves and algebraic sheaves share many structural similarities, as they both encode local-to-global properties of functions or sections
• Both types of sheaves can be defined on appropriate geometric spaces (complex manifolds for analytic sheaves, schemes for algebraic sheaves) and give rise to ringed space structures

Similarities in structure

• Analytic sheaves and algebraic sheaves both satisfy the sheaf axioms, allowing for the gluing of local sections to obtain global sections
• Both types of sheaves have notions of morphisms, kernels, cokernels, and exact sequences, enabling the study of sheaf cohomology
• Coherence and finite generation are important properties in both the analytic and algebraic settings

Key differences in properties

• Analytic sheaves are defined using holomorphic functions, while algebraic sheaves are defined using regular functions (polynomials)
• The GAGA principle (Géométrie Algébrique et Géométrie Analytique) relates analytic and algebraic sheaves on projective complex varieties, but the two categories are not equivalent in general
• Certain properties, such as the Oka-Cartan principle, hold for analytic sheaves but not for algebraic sheaves, highlighting the differences between the two settings

Analytic sheaves on complex manifolds

• Complex manifolds provide a natural setting for the study of analytic sheaves, as they are locally modeled on open subsets of $\mathbb{C}^n$ and admit a holomorphic structure
• The sheaf of holomorphic functions $\mathcal{O}_X$ on a complex manifold $X$ is the prototypical example of an analytic sheaf
• Many important results and techniques in complex analytic geometry, such as the Oka-Cartan principle and the Grauert direct image theorem, rely on the properties of analytic sheaves on complex manifolds

Existence of analytic structure

• Every complex manifold $X$ admits a natural analytic structure, given by the sheaf of holomorphic functions $\mathcal{O}_X$
• The analytic structure on $X$ is unique up to isomorphism, making it an intrinsic property of the complex manifold
• The existence of an analytic structure allows for the study of global holomorphic functions, holomorphic vector bundles, and other analytic objects on $X$

Coherent analytic sheaves

• An analytic sheaf $\mathcal{F}$ on a complex manifold $X$ is coherent if it is locally finitely generated and its stalks $\mathcal{F}_x$ are finitely generated $\mathcal{O}_{X,x}$-modules for all $x \in X$
• Coherent analytic sheaves form an abelian category, which is well-suited for the study of sheaf cohomology and other homological methods
• Examples of coherent analytic sheaves include the sheaf of holomorphic functions $\mathcal{O}_X$, the sheaf of holomorphic sections of a holomorphic vector bundle, and the ideal sheaf of a closed analytic subspace

Stein manifolds and theorems

• A complex manifold $X$ is called a Stein manifold if it satisfies certain holomorphic convexity properties, such as holomorphic separability and the existence of a strictly plurisubharmonic exhaustion function
• Stein manifolds are important in the study of analytic sheaves because they exhibit strong cohomological properties (Cartan's Theorems A and B) and have a rich function theory
• On a Stein manifold, coherent analytic sheaves are acyclic (have vanishing higher cohomology), and the global section functor is exact, providing a powerful tool for studying global holomorphic functions and sections

Cohomology of analytic sheaves

• The cohomology of analytic sheaves is a fundamental tool in complex analytic geometry, providing invariants that capture global properties of the sheaves and the underlying complex manifolds
• Several cohomology theories, such as Čech cohomology and Dolbeault cohomology, can be used to study analytic sheaves, each offering unique perspectives and computational techniques
• Sheaf cohomology plays a crucial role in classification problems, deformation theory, and the study of moduli spaces in complex analytic geometry

Čech cohomology computations

• Čech cohomology is a cohomology theory based on open covers of a topological space, which can be adapted to the study of analytic sheaves on complex manifolds
• To compute the Čech cohomology of an analytic sheaf $\mathcal{F}$ on a complex manifold $X$, one chooses a suitable open cover $\mathcal{U} = \{U_i\}$ of $X$ and considers the Čech complex associated to $\mathcal{F}$ and $\mathcal{U}$
• The Čech cohomology groups $\check{H}^p(X, \mathcal{F})$ are the cohomology groups of the Čech complex, which can be computed using algebraic methods (kernels and cokernels of the coboundary maps)

Dolbeault cohomology

• Dolbeault cohomology is a cohomology theory specific to complex manifolds, which takes into account the holomorphic structure of the manifold and the analytic sheaves
• For an analytic sheaf $\mathcal{F}$ on a complex manifold $X$, the Dolbeault cohomology groups $H^{p,q}(X, \mathcal{F})$ are defined using the Dolbeault complex, which involves the $\bar{\partial}$-operator acting on $\mathcal{F}$-valued differential forms
• Dolbeault cohomology is related to the sheaf cohomology of $\mathcal{F}$ via the Dolbeault isomorphism: $H^q(X, \Omega^p_X \otimes \mathcal{F}) \cong H^{p,q}(X, \mathcal{F})$, where $\Omega^p_X$ is the sheaf of holomorphic $p$-forms on $X$

Serre duality for analytic sheaves

• Serre duality is a fundamental result in complex analytic geometry that relates the cohomology of an analytic sheaf $\mathcal{F}$ to the cohomology of its dual sheaf $\mathcal{F}^*$, twisted by the canonical sheaf $\omega_X$
• For a compact complex manifold $X$ of dimension $n$ and a coherent analytic sheaf $\mathcal{F}$ on $X$, Serre duality states that there are natural isomorphisms: $H^q(X, \mathcal{F}) \cong H^{n-q}(X, \mathcal{F}^* \otimes \omega_X)^*$
• Serre duality provides a powerful tool for computing sheaf cohomology, as it reduces the computation of higher cohomology groups to the computation of lower cohomology groups of a related sheaf

Applications of analytic sheaves

• Analytic sheaves find numerous applications in various branches of mathematics, including complex analytic geometry, algebraic geometry, and deformation theory
• The study of analytic sheaves provides a framework for understanding the local and global properties of holomorphic functions, holomorphic vector bundles, and complex analytic spaces
• Many important results in these fields, such as the classification of complex analytic surfaces, the study of moduli spaces, and the deformation theory of complex structures, rely heavily on the theory of analytic sheaves

Complex analytic geometry

• Analytic sheaves are the building blocks of complex analytic geometry, which studies complex manifolds and analytic spaces using tools from sheaf theory and complex analysis
• The sheaf of holomorphic functions $\mathcal{O}_X$ on a complex manifold $X$ encodes the holomorphic structure of $X$ and allows for the study of global holomorphic functions and their properties
• Coherent analytic sheaves, such as holomorphic vector bundles and ideal sheaves of analytic subspaces, provide a rich source of examples and play a crucial role in the classification and study of complex analytic spaces

Deformation theory

• Deformation theory is the study of infinitesimal variations of geometric objects, such as complex manifolds, algebraic varieties, and vector bundles
• Analytic sheaves, particularly coherent analytic sheaves, are essential tools in deformation theory, as they allow for the encoding of infinitesimal deformations and the study of obstruction spaces
• The cohomology of certain analytic sheaves, such as the tangent sheaf and the normal sheaf of a submanifold, plays a central role in understanding the deformation theory of complex manifolds and their subspaces

Moduli spaces in algebraic geometry

• Moduli spaces are spaces that parameterize geometric objects, such as algebraic curves, vector bundles, or sheaves, up to certain equivalence relations
• Analytic sheaves, and their algebraic counterparts, are crucial in the construction and study of moduli spaces in algebraic geometry
• The deformation theory of coherent sheaves, which relies heavily on the properties of analytic sheaves, is a key ingredient in the construction of moduli spaces of sheaves and the study of their local and global properties