🌿Algebraic Geometry Unit 5 – Cohomology and Intersection Theory

Key Concepts and Definitions

• Sheaves generalize the notion of functions on a space by allowing local data to be attached to open sets
• Cohomology measures global properties of a sheaf that cannot be detected locally
• Čech cohomology computes sheaf cohomology using open covers and transition functions
• Cohomological dimension refers to the highest degree in which a sheaf can have non-zero cohomology
  • Spaces with low cohomological dimension have simpler global structure
• Intersection theory studies how subvarieties of an algebraic variety intersect each other
• Chow rings encode information about the intersection behavior of subvarieties
• Cycle classes represent subvarieties in the Chow ring and capture their intersection properties
• Bezout's theorem relates the intersection product of subvarieties to their degrees

Sheaf Cohomology Basics

• Sheaf cohomology extends the notion of cohomology to sheaves on a topological space
• Injective resolutions allow the computation of sheaf cohomology using derived functors
• Long exact sequence in cohomology relates the cohomology of short exact sequences of sheaves
  • Connects local and global properties of sheaves
• Higher direct images of sheaves under morphisms induce maps on cohomology
• Leray spectral sequence computes the cohomology of a sheaf using a morphism to a simpler space
• Cohomology with supports studies the cohomology of a sheaf restricted to a closed subset
• Čech-to-derived functor spectral sequence relates Čech and derived functor cohomology
• Sheaf cohomology has applications in various areas of algebraic geometry (moduli spaces, deformation theory)

Čech Cohomology and Its Applications

• Čech cohomology is a computational tool for sheaf cohomology based on open covers
• Alternating Čech cochains capture the local-to-global transition data of a sheaf
  • Cocycle condition ensures consistency of local sections
• Refinement of covers leads to isomorphic Čech cohomology groups
• Acyclicity of sheaves on certain covers allows the computation of sheaf cohomology via Čech cohomology
• Čech-to-derived functor spectral sequence relates Čech and derived functor cohomology
  • Degenerates for acyclic covers, yielding isomorphisms
• Čech cohomology is particularly useful for computing cohomology of coherent sheaves on projective varieties
• Cousin complexes provide a flasque resolution of a sheaf, enabling Čech cohomology computations
• Čech cohomology has applications in the study of line bundles, divisors, and linear systems

Cohomological Dimension and Vanishing Theorems

• Cohomological dimension measures the complexity of the global structure of a space
• Spaces with low cohomological dimension have simpler global properties
• Affine varieties have cohomological dimension equal to their Krull dimension
• Projective spaces have cohomological dimension equal to their dimension
• Künneth formula relates the cohomology of a product space to its factors
• Vanishing theorems provide conditions for the vanishing of higher cohomology groups
  • Kodaira vanishing theorem for ample line bundles on projective varieties
  • Nakano vanishing theorem for positive vector bundles on complex manifolds
• Serre duality relates the cohomology of a sheaf to its dual sheaf on a smooth projective variety
• Hodge theory studies the cohomology of smooth projective varieties over the complex numbers

Intersection Theory Fundamentals

• Intersection theory studies the intersection behavior of subvarieties in an algebraic variety
• Proper intersections occur when subvarieties meet transversely with the expected dimension
• Intersection multiplicity measures the degree of tangency between intersecting subvarieties
  • Defined using local rings or blow-ups
• Intersection product extends the intersection of subvarieties to cycle classes in the Chow ring
• Moving lemma allows the construction of proper intersections by perturbing subvarieties
• Segre classes encode the intersection behavior of a subvariety with hypersurfaces
• Todd classes measure the intersection properties of the tangent bundle of a variety
• Hirzebruch-Riemann-Roch theorem relates the Euler characteristic of a sheaf to intersection-theoretic data

Chow Rings and Cycle Classes

• Chow rings are algebraic structures that capture the intersection theory of a variety
• Cycle classes represent subvarieties in the Chow ring, encoding their intersection properties
• Rational equivalence identifies cycles that differ by the divisor of a rational function
  • Allows the construction of a well-defined intersection product
• Chow groups are the building blocks of the Chow ring, graded by codimension
• Intersection product gives the Chow ring the structure of a commutative graded ring
• Functorial properties of Chow rings under morphisms (proper pushforward, flat pullback)
• Chern classes of vector bundles live in the Chow ring and measure their intersection behavior
• Correspondence between line bundles and divisor classes in the Chow ring of a smooth variety

Intersection Products and Bezout's Theorem

• Intersection product extends the notion of intersection to cycle classes in the Chow ring
• Bezout's theorem relates the intersection product of subvarieties to their degrees
  • The degree of the intersection product equals the product of the degrees in projective space
• Multi-homogeneous Bezout's theorem for subvarieties of a product of projective spaces
• Bernstein-Kushnirenko theorem extends Bezout's theorem to the toric setting
• Intersection theory on surfaces is governed by the intersection pairing on the Picard group
• Hodge index theorem constrains the signature of the intersection pairing on a surface
• Riemann-Roch theorem for curves relates the degree and genus to the dimension of the linear system
• Excess intersection formula handles intersections of subvarieties with excessive dimension

Applications in Algebraic Geometry

• Intersection theory provides tools for computing invariants of algebraic varieties (Chern numbers, Euler characteristics)
• Schubert calculus studies the intersection theory of Grassmannians and flag varieties
  • Computes the structure constants of the Chow ring in terms of Schubert classes
• Moduli spaces of curves and their compactifications (Deligne-Mumford, Kontsevich) use intersection theory
• Gromov-Witten theory studies the intersection theory of moduli spaces of stable maps
  • Quantum cohomology encodes the intersection behavior of these moduli spaces
• Donaldson-Thomas theory investigates the intersection theory of moduli spaces of sheaves
• Intersection theory is used in the classification of algebraic varieties (Enriques-Kodaira, minimal model program)
• Resolution of singularities and the study of exceptional divisors rely on intersection-theoretic techniques
• Arakelov theory extends intersection theory to arithmetic surfaces and higher-dimensional arithmetic varieties