is a powerful tool in algebraic geometry, connecting local and global properties of varieties. It allows us to study , , and important invariants like and .
This section explores applications of sheaf cohomology in algebraic geometry. We'll see how it's used to classify vector bundles, compute invariants of varieties, and relate to other cohomology theories like Čech, singular, étale, and .
Sheaf cohomology for coherent sheaves
Properties and applications of coherent sheaves
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Coherent sheaves are sheaves of modules over the structure sheaf of an algebraic variety that are locally finitely generated and locally finitely presented
This means they can be described locally by a finite number of generators and relations
Examples of coherent sheaves include the structure sheaf itself and sheaves of sections of vector bundles
Sheaf cohomology provides a powerful tool to study global sections and higher cohomology groups of coherent sheaves on algebraic varieties
The zeroth cohomology group H0(X,F) represents the global sections of the sheaf F
Higher cohomology groups Hi(X,F) measure the obstruction to extending local sections globally
The dimension of the cohomology groups of a coherent sheaf can be used to determine properties such as the rank, degree, and of the sheaf
The rank of a coherent sheaf is the dimension of the stalk at a generic point (the fiber of the sheaf over that point)
The degree of a coherent sheaf on a projective variety is the degree of the corresponding cycle in the Chow ring
The Euler characteristic χ(X,F) is the alternating sum of the dimensions of the cohomology groups
Vanishing theorems and duality
Vanishing theorems, such as the and the , provide conditions under which certain cohomology groups of coherent sheaves vanish
The Kodaira vanishing theorem states that for an ample line bundle L on a smooth projective variety X over C, Hi(X,KX⊗L)=0 for i>0, where KX is the
The Serre vanishing theorem states that for a coherent sheaf F on a projective variety X and a sufficiently ample line bundle L, Hi(X,F⊗L)=0 for i>0
relates the cohomology groups of a coherent sheaf to the cohomology groups of its dual sheaf, providing a powerful tool for computing cohomology
For a coherent sheaf F on a smooth projective variety X of dimension n over a field k, there are isomorphisms Hi(X,F)≅Hn−i(X,F∨⊗ωX)∨, where F∨ is the dual sheaf and ωX is the canonical sheaf
This allows the computation of cohomology groups by reducing to the dual sheaf and using vanishing theorems
Sheaf cohomology in vector bundle classification
Vector bundles and the Picard group
Vector bundles are locally free sheaves, which means they are locally isomorphic to a direct sum of copies of the structure sheaf
A rank r vector bundle on a variety X is a sheaf E that is locally isomorphic to OX⊕r
Line bundles are vector bundles of rank 1 and play a crucial role in the classification of varieties
The set of isomorphism classes of vector bundles on an algebraic variety forms an abelian group called the , which can be studied using sheaf cohomology
The Picard group Pic(X) is the group of isomorphism classes of line bundles on X with the tensor product operation
There is an isomorphism Pic(X)≅H1(X,OX×), where OX× is the sheaf of invertible functions on X
Chern classes and the Riemann-Roch theorem
The of a vector bundle is an element of the second cohomology group of the variety with coefficients in the sheaf of invertible functions, and it provides an important invariant for classifying vector bundles
The first Chern class c1(E) of a vector bundle E is an element of H2(X,OX×) that measures the obstruction to the existence of a global trivialization of E
The first Chern class of a line bundle L is the image of the isomorphism class of L under the isomorphism Pic(X)≅H1(X,OX×)
Extensions of vector bundles are classified by the first cohomology group of the sheaf of homomorphisms between the bundles, which can be computed using sheaf cohomology
An extension of vector bundles 0→E′→E→E′′→0 is classified by an element of H1(X,Hom(E′′,E′))
The sheaf of homomorphisms Hom(E′′,E′) can be studied using sheaf cohomology to determine the possible extensions
The relates the Euler characteristic of a vector bundle to its Chern classes, providing a powerful tool for studying the geometry of vector bundles
For a vector bundle E on a smooth projective variety X of dimension n, the Riemann-Roch theorem states that χ(X,E)=∫Xch(E)⋅td(X), where ch(E) is the Chern character of E and td(X) is the Todd class of X
The Chern character and Todd class are expressed in terms of Chern classes and can be computed using sheaf cohomology
Sheaf cohomology for algebraic variety invariants
Cohomology of the structure sheaf and Hodge numbers
The cohomology groups of the structure sheaf of an algebraic variety provide important invariants, such as the dimension, genus, and of the variety
The dimension of a variety X is the largest integer n such that Hn(X,OX)=0
The genus of a smooth projective curve C is g=dimH1(C,OC)
The arithmetic genus of a variety X is pa(X)=(−1)dimX(χ(X,OX)−1)
The Hodge numbers of a smooth projective variety can be computed using the Hodge decomposition of the cohomology groups of the sheaf of differential forms
The Hodge numbers hp,q(X) of a smooth projective variety X are the dimensions of the cohomology groups Hq(X,ΩXp), where ΩXp is the sheaf of holomorphic p-forms on X
The Hodge decomposition states that Hk(X,C)≅⨁p+q=kHq(X,ΩXp), allowing the computation of Hodge numbers using sheaf cohomology
Canonical bundle and Kodaira dimension
The canonical bundle of a variety, which is the determinant of the cotangent bundle, plays a crucial role in the classification of varieties and can be studied using sheaf cohomology
The canonical bundle ωX of a smooth variety X is the determinant of the cotangent bundle ΩX1, i.e., ωX=detΩX1
The pluricanonical bundles ωX⊗k are tensor powers of the canonical bundle and their cohomology groups provide important invariants
The of a variety, which measures the growth of pluricanonical forms, can be computed using the dimensions of the cohomology groups of the pluricanonical bundles
The Kodaira dimension κ(X) of a variety X is the maximum dimension of the image of X under the rational maps defined by the pluricanonical linear systems ∣mωX∣ for sufficiently divisible m>0, or −∞ if all pluricanonical linear systems are empty
The Kodaira dimension can be computed using the asymptotic behavior of the dimensions of the cohomology groups H0(X,ωX⊗m) as m→∞
Sheaf cohomology vs other cohomology theories
Čech and singular cohomology
Sheaf cohomology is related to , which is defined using open covers of a variety and provides a more explicit way to compute cohomology groups
Čech cohomology Hˇi(X,F) of a sheaf F on a variety X is defined using an open cover U of X and the Čech complex Cˇ∙(U,F)
For a sufficiently fine open cover, sheaf cohomology is isomorphic to Čech cohomology, i.e., Hi(X,F)≅Hˇi(X,F)
For smooth varieties, sheaf cohomology is isomorphic to , which is defined using singular chains and provides a topological perspective on cohomology
Singular cohomology Hi(X,Z) of a topological space X is defined using the dual of the singular chain complex, which is constructed using continuous maps from simplices to X
For a smooth variety X over C, there are isomorphisms Hi(X,Z)⊗C≅Hi(X,C)≅⨁p+q=iHq(X,ΩXp), relating singular cohomology to sheaf cohomology and Hodge theory
Étale and crystalline cohomology
is a cohomology theory for algebraic varieties that takes into account the arithmetic properties of the variety and is related to sheaf cohomology through the étale topology
Étale cohomology Heˊti(X,F) of a sheaf F on a variety X is defined using the étale site of X, which is a Grothendieck topology that captures the arithmetic properties of X
For a smooth proper variety X over a field k, there are comparison theorems relating étale cohomology to sheaf cohomology, such as the isomorphism Heˊti(X,Zℓ)⊗Qℓ≅Hi(X,Qℓ) for ℓ=char(k)
is a p-adic cohomology theory that is related to sheaf cohomology through the theory of crystals and provides a way to study varieties over fields of positive characteristic
Crystalline cohomology Hcrisi(X/W) of a smooth proper variety X over a perfect field k of characteristic p>0 is defined using the crystalline site of X over the ring of Witt vectors W=W(k)
There are comparison theorems relating crystalline cohomology to other cohomology theories, such as the de Rham-Witt complex and the Hodge-Witt cohomology, which are analogues of de Rham cohomology and Hodge theory in positive characteristic
De Rham cohomology
De Rham cohomology, which is defined using differential forms, is isomorphic to sheaf cohomology for smooth varieties over fields of characteristic zero
De Rham cohomology HdRi(X) of a smooth variety X over a field k of characteristic zero is defined as the hypercohomology of the de Rham complex ΩX∙
The algebraic de Rham theorem states that there is an isomorphism HdRi(X)≅Hi(X,C) for a smooth variety X over C, relating de Rham cohomology to sheaf cohomology
The Hodge filtration on de Rham cohomology induces the Hodge decomposition on the cohomology groups of the sheaf of differential forms, providing a connection to Hodge theory