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
Key Terms to Review (20)
Arithmetic genus: The arithmetic genus is a topological invariant that measures the geometric complexity of a projective variety, specifically reflecting the number of independent meromorphic differentials on it. This concept is crucial in algebraic geometry, where it helps to classify and understand the properties of algebraic curves and surfaces. The arithmetic genus is related to other invariants like the geometric genus and plays a significant role in the study of Riemann surfaces and their applications in complex geometry.
Canonical bundle: A canonical bundle is a line bundle associated with a smooth projective variety, which captures essential geometric and topological information about the variety. This bundle is defined using the sheaf of differentials, allowing mathematicians to study properties like divisors, sections, and cohomology. The canonical bundle is significant in algebraic geometry as it provides a way to link the geometric structure of varieties to their algebraic properties.
čech cohomology: Čech cohomology is a type of cohomology theory used in algebraic topology that provides a way to associate algebraic invariants to topological spaces. It is defined using open covers of a space and focuses on the relationships between the local data provided by these open sets. This concept connects deeply with various aspects of topology, including its axiomatic foundations, applications in algebraic geometry, and the broader study of homological properties of spaces.
Chern classes: Chern classes are a type of characteristic class that provide important topological invariants for complex vector bundles. They capture the geometry of a vector bundle and help to classify them up to isomorphism, linking algebraic properties with topological features. These classes have significant implications in various mathematical fields, influencing concepts related to fiber bundles, vector bundles, and even applications in algebraic geometry.
Coherent Sheaves: Coherent sheaves are a special type of sheaf that captures the idea of locally finitely generated modules over a ring. They are essential in algebraic geometry as they allow for the study of geometric objects through their algebraic properties, providing a way to relate local data to global structures. Coherent sheaves facilitate the analysis of varieties by enabling the use of tools from commutative algebra and homological algebra.
Crystalline cohomology: Crystalline cohomology is a powerful tool in algebraic geometry that provides a way to study schemes over a field of positive characteristic, particularly in relation to their geometry and arithmetic. It is designed to capture more geometric information than traditional cohomology theories by focusing on the behavior of schemes in a 'crystalline' context, allowing for connections to be made between algebraic properties and topological features. This theory plays an essential role in understanding the structure of algebraic varieties and their interactions with various mathematical objects.
De Rham cohomology: de Rham cohomology is a tool in algebraic topology that associates a sequence of cohomology groups to a smooth manifold, capturing information about the manifold's differential forms. It connects deeply with other cohomology theories, such as Čech cohomology, and provides insights into geometric and topological properties through the study of closed and exact forms.
étale cohomology: Étale cohomology is a powerful tool in algebraic geometry that provides a way to study the topology of algebraic varieties over fields, particularly finite fields, using techniques from both algebra and topology. It generalizes the notion of sheaf cohomology to the étale topology, which allows for a finer analysis of geometric properties. This framework connects algebraic geometry with number theory and has applications in various areas such as arithmetic geometry and the study of Galois representations.
Euler Characteristic: The Euler characteristic is a topological invariant that represents a fundamental property of a space, defined as the alternating sum of the number of vertices, edges, and faces in a polyhedron, given by the formula $$ ext{χ} = V - E + F$$. This invariant helps classify surfaces and can also extend to higher-dimensional spaces through more complex definitions. It connects various concepts such as homology, duality, and manifold characteristics, making it essential in understanding topological properties and relationships.
First Chern Class: The first Chern class is a topological invariant associated with complex vector bundles, serving as a measure of their curvature. It plays a crucial role in algebraic geometry by providing a way to link the geometry of a variety with its topology, particularly through the use of line bundles. The first Chern class helps in understanding how these bundles can be classified and the relationship between the geometry of algebraic varieties and their cohomological properties.
Hodge numbers: Hodge numbers are integers that arise in the study of algebraic geometry and algebraic topology, representing the dimensions of certain cohomology groups of a non-singular projective variety. They provide crucial information about the geometric structure of the variety, linking its topology and complex geometry through the Hodge decomposition theorem. Hodge numbers help classify varieties and are fundamental in understanding their geometric properties, such as their shape and symmetry.
Kodaira Dimension: Kodaira dimension is a fundamental concept in algebraic geometry that measures the growth of the space of global sections of line bundles on a projective variety. It classifies varieties based on their geometric properties and the behavior of their canonical sheaves, helping to distinguish between different types of varieties such as Fano, Calabi-Yau, and general type.
Kodaira Vanishing Theorem: The Kodaira Vanishing Theorem states that for a smooth projective variety over a field, the higher cohomology groups of certain line bundles vanish under specific conditions. This result is crucial in algebraic geometry as it establishes important connections between the geometry of a variety and its cohomological properties, leading to various applications in the study of complex manifolds and algebraic varieties.
Picard group: The Picard group is an important concept in algebraic geometry that refers to the group of line bundles (or more generally, invertible sheaves) on a given algebraic variety, with group operation defined by tensor product. This group plays a significant role in understanding the geometry of varieties and their divisor classes, helping to connect algebraic geometry with topology through the study of morphisms and coherent sheaves.
Riemann-Roch Theorem: The Riemann-Roch Theorem is a fundamental result in algebraic geometry and complex analysis that relates the geometry of a compact Riemann surface to the algebraic properties of line bundles on that surface. It provides a powerful way to calculate dimensions of spaces of meromorphic functions and differentials, linking topological data, like the genus of a surface, to analytical properties, which has broad implications in various mathematical fields.
Serre Duality: Serre Duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of a topological space. It establishes an important duality between the k-th homology group and the (n-k)-th cohomology group of a compact, orientable manifold, where n is the dimension of the manifold. This duality has significant implications in algebraic geometry, particularly in understanding the relationships between different types of cohomological invariants and their geometric interpretations.
Serre Vanishing Theorem: The Serre Vanishing Theorem states that for a coherent sheaf on a projective variety, there exists an integer $n$ such that the sheaf becomes globally generated when restricted to the n-th twist. This theorem is important in algebraic geometry as it provides a powerful tool for understanding the behavior of sheaves on projective varieties and helps link cohomological properties with geometric properties.
Sheaf Cohomology: Sheaf cohomology is a mathematical tool used to study the global properties of sheaves, which are data assignments to open sets of a topological space, often capturing local information that can be extended globally. It connects the concepts of topology and algebra by allowing for the computation of derived functors, particularly in understanding how local sections of sheaves can be patched together to yield global sections. This method is essential in various fields, including algebraic geometry, where it helps analyze the properties of varieties and their associated sheaves.
Singular cohomology: Singular cohomology is a mathematical tool in algebraic topology that assigns a sequence of abelian groups or vector spaces to a topological space, providing a way to study its shape and structure. This concept extends the idea of singular homology by incorporating the duality of the spaces through the use of cochains, allowing for a deeper analysis of topological properties. It plays a crucial role in connecting various mathematical disciplines, including differential geometry and algebraic geometry, while adhering to the foundational axioms that define cohomology theories.
Vector Bundles: A vector bundle is a mathematical structure that consists of a topological space, called the base space, and a vector space attached to each point in that space, providing a way to study the properties of these spaces through linear algebra. This concept helps in understanding how vector spaces vary continuously over a manifold, allowing us to link geometry with algebraic concepts. Vector bundles are crucial in various fields, connecting topology with cohomology theories, manifold theory, and even applications in algebraic geometry.