🍃Sheaf Theory Unit 8 – Sheaves in algebraic geometry

Key Concepts and Definitions

• Sheaves generalize the notion of functions on a topological space by assigning data to open sets in a way that is compatible with restrictions
• Presheaves consist of a contravariant functor from the category of open sets of a topological space $X$ to a category $\mathcal{C}$, often the category of sets, abelian groups, or rings
  • Morphisms between presheaves are natural transformations between the corresponding functors
• Sheaves satisfy the gluing axiom, which states that sections on overlapping open sets can be uniquely glued together when they agree on the intersection
• Stalks of a sheaf $\mathcal{F}$ at a point $x \in X$ are defined as the direct limit of $\mathcal{F}(U)$ over all open neighborhoods $U$ of $x$
• Sheafification is the process of turning a presheaf into a sheaf by adding the gluing property
• Sheaf morphisms are morphisms of presheaves that respect the sheaf structure
• Exact sequences of sheaves play a crucial role in studying the local and global properties of sheaves

Historical Context and Development

• The concept of sheaves originated in the 1940s through the work of Jean Leray, who introduced them as a tool to study the topology of spaces using cohomology
• Henri Cartan and Jean-Pierre Serre further developed sheaf theory in the 1950s, establishing its foundations and applying it to complex analytic geometry
• Alexander Grothendieck revolutionized algebraic geometry in the 1960s by introducing schemes and the systematic use of sheaves, leading to a new framework for the field
  • Grothendieck's approach allowed for the unification of various branches of geometry and the development of powerful cohomological techniques
• Pierre Deligne, Luc Illusie, and others continued to expand and refine the theory of sheaves, proving important results such as the Weil conjectures and the Hodge theory for singular varieties
• The étale topology, introduced by Grothendieck and Michael Artin, provided a new perspective on sheaves and cohomology, leading to significant advances in arithmetic geometry

Sheaves in Algebraic Geometry

• In algebraic geometry, sheaves are used to study the local and global properties of algebraic varieties and schemes
• The structure sheaf $\mathcal{O}_X$ of a scheme $X$ encodes the algebraic structure of $X$ by assigning to each open set the ring of regular functions on that set
  • Morphisms between schemes induce morphisms between their structure sheaves
• Quasi-coherent sheaves generalize the notion of modules over a ring to the setting of schemes, with the structure sheaf acting as the base ring
  • Coherent sheaves are quasi-coherent sheaves that are locally finitely generated
• Vector bundles on a scheme $X$ correspond to locally free sheaves of $\mathcal{O}_X$-modules
• Invertible sheaves, or line bundles, are rank-one locally free sheaves that play a central role in the study of divisors and the Picard group
• The canonical sheaf $\omega_X$ of a smooth variety $X$ is the sheaf of top differential forms, which is closely related to the dualizing sheaf in the singular case

Construction and Properties

• Sheaves can be constructed from presheaves by sheafification, which involves adding sections to ensure the gluing property holds
• The category of sheaves on a topological space $X$ is an abelian category, allowing for the use of homological algebra techniques
  • The kernel, cokernel, and image of sheaf morphisms are defined pointwise on open sets
• Sheaf Hom, denoted $\mathcal{H}om(\mathcal{F}, \mathcal{G})$, is the sheaf of morphisms between two sheaves $\mathcal{F}$ and $\mathcal{G}$
  • The global sections of $\mathcal{H}om(\mathcal{F}, \mathcal{G})$ correspond to morphisms between $\mathcal{F}$ and $\mathcal{G}$
• Tensor products of sheaves, denoted $\mathcal{F} \otimes \mathcal{G}$, can be defined using the tensor product of the underlying categories (e.g., modules or abelian groups)
• Pullbacks and pushforwards of sheaves along continuous maps or morphisms of schemes allow for the study of sheaves under base change
• The support of a sheaf $\mathcal{F}$ is the set of points $x \in X$ where the stalk $\mathcal{F}_x$ is non-zero, providing a notion of the "location" of the sheaf

Sheaf Cohomology

• Sheaf cohomology is a powerful tool for studying the global properties of sheaves and the underlying topological space or scheme
• The cohomology groups $H^i(X, \mathcal{F})$ measure the obstructions to solving certain local-to-global problems related to the sheaf $\mathcal{F}$ on the space $X$
  • For example, $H^1(X, \mathcal{O}_X)$ classifies line bundles on $X$ up to isomorphism
• Čech cohomology provides a concrete way to compute sheaf cohomology using open covers and Čech cocycles
• The long exact sequence in cohomology relates the cohomology of a short exact sequence of sheaves, allowing for inductive computations and comparisons
• Serre duality relates the cohomology of a coherent sheaf on a smooth projective variety to the cohomology of its dual sheaf twisted by the canonical bundle
• Higher direct images $R^if_*\mathcal{F}$ and the Leray spectral sequence relate the cohomology of a sheaf on a space to its cohomology on the base of a morphism

Applications in Algebraic Geometry

• Sheaf cohomology is used to study the geometry of algebraic varieties and schemes, providing invariants and obstructions
• The Picard group $\text{Pic}(X)$, classifying line bundles on $X$, can be identified with $H^1(X, \mathcal{O}_X^*)$, where $\mathcal{O}_X^*$ is the sheaf of units
• The canonical bundle and its cohomology play a central role in the classification of algebraic surfaces and the study of minimal models
• The Riemann-Roch theorem relates the Euler characteristic of a coherent sheaf to intersection-theoretic data, providing a powerful tool for computations
• Serre's GAGA principle (Géométrie Algébrique et Géométrie Analytique) establishes a correspondence between algebraic and analytic coherent sheaves on complex projective varieties
• Sheaves and their cohomology are essential in the study of moduli spaces, which parametrize geometric objects such as curves, surfaces, or vector bundles

Related Theories and Connections

• Derived categories and derived functors provide a more general framework for studying sheaves and their cohomology, allowing for the use of resolutions and derived equivalences
• Perverse sheaves, which arise in the study of intersection cohomology and the decomposition theorem, are a special class of sheaves with good properties under Verdier duality
• D-modules, which are sheaves with an action of differential operators, play a key role in the Riemann-Hilbert correspondence and the study of linear partial differential equations
• Hodge theory and mixed Hodge modules relate sheaf cohomology to the Hodge structure on the cohomology of complex algebraic varieties
• Motivic sheaves and motives aim to provide a unified framework for the study of cohomology theories in algebraic geometry, including étale cohomology and Hodge theory
• Sheaf theory has connections to other areas of mathematics, such as representation theory (via perverse sheaves and the geometric Langlands program), mathematical physics (via twistor theory and instantons), and topology (via constructible sheaves and intersection homology)

Challenges and Advanced Topics

• Computing sheaf cohomology explicitly can be difficult, especially for singular or non-projective varieties, requiring the use of resolutions and spectral sequences
• The study of perverse sheaves and the decomposition theorem involves deep results in algebraic geometry and representation theory, such as the Kazhdan-Lusztig conjecture and the Riemann-Hilbert correspondence
• The theory of D-modules and its applications to representation theory and mathematical physics require a significant amount of technical machinery, including the Riemann-Hilbert correspondence and the theory of holonomic D-modules
• The geometric Langlands program, which relates sheaves on moduli spaces of bundles to representations of reductive groups, is a vast and active area of research with connections to number theory, representation theory, and mathematical physics
• The study of motives and motivic sheaves aims to provide a universal cohomology theory for algebraic varieties, but the existence of a suitable category of motives is still conjectural and the subject of ongoing research
• The theory of mixed Hodge modules, which combines Hodge theory and perverse sheaves, involves intricate constructions and deep results in algebraic geometry and Hodge theory, such as the decomposition theorem and the theory of variations of Hodge structure
• The application of sheaf theory to arithmetic geometry, such as the study of étale cohomology and the Weil conjectures, requires a thorough understanding of the étale topology and the machinery of algebraic geometry in positive characteristic