11.3 Applications to algebraic geometry

Sheaf cohomology is a powerful tool in algebraic geometry, connecting local and global properties of varieties. It allows us to study coherent sheaves, vector bundles, and important invariants like Chern classes and Hodge numbers.

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 de Rham cohomology.

Sheaf cohomology for coherent sheaves

Properties and applications of coherent sheaves

• 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 $H^0(X, \mathcal{F})$ represents the global sections of the sheaf $\mathcal{F}$
  • Higher cohomology groups $H^i(X, \mathcal{F})$ 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 Euler characteristic 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 $\chi(X, \mathcal{F})$ is the alternating sum of the dimensions of the cohomology groups

Vanishing theorems and duality

• Vanishing theorems, such as the Kodaira vanishing theorem and the Serre vanishing theorem, 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 $\mathbb{C}$, $H^i(X, K_X \otimes L) = 0$ for $i > 0$, where $K_X$ is the canonical bundle
  • The Serre vanishing theorem states that for a coherent sheaf $\mathcal{F}$ on a projective variety $X$ and a sufficiently ample line bundle $L$, $H^i(X, \mathcal{F} \otimes L) = 0$ for $i > 0$
• Serre duality 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 $\mathcal{F}$ on a smooth projective variety $X$ of dimension $n$ over a field $k$, there are isomorphisms $H^i(X, \mathcal{F}) \cong H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)^\vee$, where $\mathcal{F}^\vee$ is the dual sheaf and $\omega_X$ 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 $\mathcal{E}$ that is locally isomorphic to $\mathcal{O}_X^{\oplus r}$
  • 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 Picard group, which can be studied using sheaf cohomology
  • The Picard group $\text{Pic}(X)$ is the group of isomorphism classes of line bundles on $X$ with the tensor product operation
  • There is an isomorphism $\text{Pic}(X) \cong H^1(X, \mathcal{O}_X^\times)$, where $\mathcal{O}_X^\times$ is the sheaf of invertible functions on $X$

Chern classes and the Riemann-Roch theorem

• The first Chern class 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 $c_1(\mathcal{E})$ of a vector bundle $\mathcal{E}$ is an element of $H^2(X, \mathcal{O}_X^\times)$ that measures the obstruction to the existence of a global trivialization of $\mathcal{E}$
  • The first Chern class of a line bundle $L$ is the image of the isomorphism class of $L$ under the isomorphism $\text{Pic}(X) \cong H^1(X, \mathcal{O}_X^\times)$
• 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 $0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}'' \to 0$ is classified by an element of $H^1(X, \mathcal{H}om(\mathcal{E}'', \mathcal{E}'))$
  • The sheaf of homomorphisms $\mathcal{H}om(\mathcal{E}'', \mathcal{E}')$ can be studied using sheaf cohomology to determine the possible extensions
• The Riemann-Roch theorem 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 $\mathcal{E}$ on a smooth projective variety $X$ of dimension $n$, the Riemann-Roch theorem states that $\chi(X, \mathcal{E}) = \int_X \text{ch}(\mathcal{E}) \cdot \text{td}(X)$, where $\text{ch}(\mathcal{E})$ is the Chern character of $\mathcal{E}$ and $\text{td}(X)$ 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 arithmetic genus of the variety
  • The dimension of a variety $X$ is the largest integer $n$ such that $H^n(X, \mathcal{O}_X) \neq 0$
  • The genus of a smooth projective curve $C$ is $g = \dim H^1(C, \mathcal{O}_C)$
  • The arithmetic genus of a variety $X$ is $p_a(X) = (-1)^{\dim X}(\chi(X, \mathcal{O}_X) - 1)$
• 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 $h^{p,q}(X)$ of a smooth projective variety $X$ are the dimensions of the cohomology groups $H^q(X, \Omega_X^p)$, where $\Omega_X^p$ is the sheaf of holomorphic $p$-forms on $X$
  • The Hodge decomposition states that $H^k(X, \mathbb{C}) \cong \bigoplus_{p+q=k} H^q(X, \Omega_X^p)$, 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 $\omega_X$ of a smooth variety $X$ is the determinant of the cotangent bundle $\Omega_X^1$, i.e., $\omega_X = \det \Omega_X^1$
  • The pluricanonical bundles $\omega_X^{\otimes k}$ are tensor powers of the canonical bundle and their cohomology groups provide important invariants
• The Kodaira dimension 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 $\kappa(X)$ of a variety $X$ is the maximum dimension of the image of $X$ under the rational maps defined by the pluricanonical linear systems $|m\omega_X|$ for sufficiently divisible $m > 0$, or $-\infty$ if all pluricanonical linear systems are empty
  • The Kodaira dimension can be computed using the asymptotic behavior of the dimensions of the cohomology groups $H^0(X, \omega_X^{\otimes m})$ as $m \to \infty$

Sheaf cohomology vs other cohomology theories

Čech and singular cohomology

• Sheaf cohomology is related to Čech cohomology, which is defined using open covers of a variety and provides a more explicit way to compute cohomology groups
  • Čech cohomology $\check{H}^i(X, \mathcal{F})$ of a sheaf $\mathcal{F}$ on a variety $X$ is defined using an open cover $\mathfrak{U}$ of $X$ and the Čech complex $\check{C}^\bullet(\mathfrak{U}, \mathcal{F})$
  • For a sufficiently fine open cover, sheaf cohomology is isomorphic to Čech cohomology, i.e., $H^i(X, \mathcal{F}) \cong \check{H}^i(X, \mathcal{F})$
• For smooth varieties, sheaf cohomology is isomorphic to singular cohomology, which is defined using singular chains and provides a topological perspective on cohomology
  • Singular cohomology $H^i(X, \mathbb{Z})$ 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 $\mathbb{C}$, there are isomorphisms $H^i(X, \mathbb{Z}) \otimes \mathbb{C} \cong H^i(X, \mathbb{C}) \cong \bigoplus_{p+q=i} H^q(X, \Omega_X^p)$, relating singular cohomology to sheaf cohomology and Hodge theory

Étale and crystalline cohomology

• Étale 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 $H^i_{\text{ét}}(X, \mathcal{F})$ of a sheaf $\mathcal{F}$ 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 $H^i_{\text{ét}}(X, \mathbb{Z}_\ell) \otimes \mathbb{Q}_\ell \cong H^i(X, \mathbb{Q}_\ell)$ for $\ell \neq \text{char}(k)$
• Crystalline cohomology 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 $H^i_{\text{cris}}(X/W)$ 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 $H^i_{\text{dR}}(X)$ of a smooth variety $X$ over a field $k$ of characteristic zero is defined as the hypercohomology of the de Rham complex $\Omega_X^\bullet$
  • The algebraic de Rham theorem states that there is an isomorphism $H^i_{\text{dR}}(X) \cong H^i(X, \mathbb{C})$ for a smooth variety $X$ over $\mathbb{C}$, 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