9.2 Chern classes

Chern classes are crucial invariants in algebraic topology and differential geometry. They measure the twisting of complex vector bundles over topological spaces, providing insights into the geometry and topology of the base space.

These classes satisfy key axioms like naturality and the Whitney sum formula. They're universal characteristic classes for complex vector bundles, making them fundamental in studying bundle invariants and related mathematical structures.

Definition of Chern classes

• Chern classes are important invariants in algebraic topology and differential geometry that assign cohomology classes to complex vector bundles over a topological space or manifold
• They provide a way to measure the twisting and non-triviality of vector bundles, which is crucial in understanding the geometry and topology of the base space

Chern classes for complex vector bundles

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• Let $E \to X$ be a complex vector bundle of rank $r$ over a topological space $X$
• The $i$-th Chern class of $E$, denoted by $c_i(E)$, is an element of the cohomology group $H^{2i}(X; \mathbb{Z})$ for $i = 0, 1, \ldots, r$
• The total Chern class is defined as the formal sum $c(E) = 1 + c_1(E) + \ldots + c_r(E)$

Axiomatic characterization of Chern classes

• Chern classes satisfy the following axioms:
  1. Naturality: For any continuous map $f: Y \to X$ and any complex vector bundle $E \to X$, $f^*(c_i(E)) = c_i(f^*E)$, where $f^*E$ is the pullback bundle
  2. Normalization: For the tautological line bundle $\gamma^1 \to \mathbb{CP}^1$, $c_1(\gamma^1) = -1 \in H^2(\mathbb{CP}^1; \mathbb{Z}) \cong \mathbb{Z}$
  3. Whitney sum formula: For any two complex vector bundles $E$ and $F$ over $X$, $c(E \oplus F) = c(E) \cdot c(F)$
• These axioms uniquely determine the Chern classes for any complex vector bundle

Universal property of Chern classes

• The Chern classes are universal characteristic classes for complex vector bundles
• Any other characteristic class for complex vector bundles can be expressed as a polynomial in the Chern classes
• This universality property makes Chern classes a fundamental tool in the study of complex vector bundles and their associated invariants

Properties of Chern classes

• Chern classes possess several important properties that make them useful in various mathematical contexts
• These properties allow for the computation and manipulation of Chern classes in different situations

Functoriality of Chern classes

• Chern classes are functorial with respect to pullbacks and isomorphisms of vector bundles
• If $f: Y \to X$ is a continuous map and $E \to X$ is a complex vector bundle, then $f^*(c_i(E)) = c_i(f^*E)$
• If $E$ and $F$ are isomorphic vector bundles over $X$, then $c_i(E) = c_i(F)$ for all $i$

Whitney sum formula for Chern classes

• The Whitney sum formula relates the Chern classes of a direct sum of vector bundles to the Chern classes of the individual bundles
• For any two complex vector bundles $E$ and $F$ over $X$, $c(E \oplus F) = c(E) \cdot c(F)$
• This formula allows for the computation of Chern classes of direct sums in terms of the Chern classes of the summands

Chern classes of tensor products

• There is a formula for the Chern classes of the tensor product of two vector bundles
• If $E$ and $F$ are complex vector bundles over $X$ of ranks $r$ and $s$ respectively, then $c(E \otimes F) = \prod_{i=1}^r \prod_{j=1}^s (1 + x_i + y_j)$, where $c(E) = \prod_{i=1}^r (1 + x_i)$ and $c(F) = \prod_{j=1}^s (1 + y_j)$
• This formula expresses the Chern classes of the tensor product in terms of the Chern classes of the factors

Chern classes of dual bundles

• The Chern classes of the dual bundle $E^*$ of a complex vector bundle $E$ are related to the Chern classes of $E$
• If $E$ has rank $r$, then $c_i(E^*) = (-1)^i c_i(E)$ for all $i$
• This property allows for the computation of Chern classes of dual bundles from the Chern classes of the original bundle

Chern classes and pullbacks

• Chern classes are compatible with pullbacks of vector bundles
• If $f: Y \to X$ is a continuous map and $E \to X$ is a complex vector bundle, then $f^*(c_i(E)) = c_i(f^*E)$, where $f^*E$ is the pullback bundle over $Y$
• This property is crucial in studying the behavior of Chern classes under maps between spaces

Computation of Chern classes

• Computing Chern classes is an important task in algebraic topology and geometry
• Several techniques and results are available for calculating Chern classes in various situations

Chern classes of line bundles

• For a complex line bundle $L \to X$, the only non-trivial Chern class is the first Chern class $c_1(L) \in H^2(X; \mathbb{Z})$
• The first Chern class of a line bundle is the Euler class of the associated oriented real vector bundle
• In the case of the tautological line bundle $\gamma^1 \to \mathbb{CP}^1$, $c_1(\gamma^1)$ generates $H^2(\mathbb{CP}^1; \mathbb{Z}) \cong \mathbb{Z}$

Splitting principle for Chern classes

• The splitting principle is a technique for reducing the computation of Chern classes of a vector bundle to the case of line bundles
• It states that for any complex vector bundle $E \to X$ of rank $r$, there exists a space $F(E)$ and a map $\pi: F(E) \to X$ such that $\pi^*E$ splits as a direct sum of line bundles and the induced map $\pi^*: H^*(X; \mathbb{Z}) \to H^*(F(E); \mathbb{Z})$ is injective
• The Chern classes of $E$ can then be computed using the Chern classes of the line bundles in the splitting

Chern classes of projective space bundles

• For a complex vector bundle $E \to X$ of rank $r$, the projectivization $\mathbb{P}(E)$ is a fiber bundle over $X$ with fiber $\mathbb{CP}^{r-1}$
• The cohomology ring of $\mathbb{P}(E)$ can be described in terms of the Chern classes of $E$ and the first Chern class of the tautological line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$
• Specifically, $H^*(\mathbb{P}(E); \mathbb{Z}) \cong H^*(X; \mathbb{Z})[t] / (t^r + c_1(E)t^{r-1} + \ldots + c_r(E))$, where $t = c_1(\mathcal{O}_{\mathbb{P}(E)}(1))$

Chern character and Chern classes

• The Chern character is a ring homomorphism from the K-theory of a space $X$ to its rational cohomology
• For a complex vector bundle $E \to X$, the Chern character is defined as $ch(E) = \sum_{i=0}^\infty \frac{1}{i!}c_1(E)^i$
• The Chern character and the Chern classes are related by the Newton identities, which express the Chern classes in terms of the Chern character

Applications of Chern classes

• Chern classes have numerous applications in various branches of mathematics, including topology, geometry, and mathematical physics
• They provide important invariants and tools for studying the properties of vector bundles and their base spaces

Chern classes and characteristic classes

• Chern classes are a special case of characteristic classes, which are cohomology classes associated with principal bundles and their associated vector bundles
• Other examples of characteristic classes include Stiefel-Whitney classes for real vector bundles and Pontryagin classes for real vector bundles with an orientation
• Chern classes are the primary characteristic classes for complex vector bundles and play a central role in the theory of characteristic classes

Chern classes and obstruction theory

• Chern classes can be used to study the existence and uniqueness of certain geometric structures on manifolds
• For example, the vanishing of certain Chern classes is a necessary condition for a complex manifold to admit a Kähler metric
• In general, Chern classes provide obstructions to the existence of various structures, such as almost complex structures, holomorphic sections, and conformal immersions

Chern classes and divisors

• In complex algebraic geometry, the first Chern class of a line bundle is closely related to the concept of a divisor
• A divisor on a complex manifold $X$ is a formal linear combination of codimension-one subvarieties (hypersurfaces) of $X$
• The first Chern class of a line bundle $L \to X$ can be represented by the divisor of any meromorphic section of $L$
• This correspondence between line bundles and divisors is a fundamental tool in the study of complex algebraic varieties

Chern classes and Chern numbers

• Chern numbers are invariants obtained by evaluating certain polynomials in the Chern classes on the fundamental class of a compact complex manifold
• For a compact complex manifold $X$ of dimension $n$, the Chern numbers are defined as $\int_X P(c_1(TX), \ldots, c_n(TX))$, where $P$ is a polynomial of degree $n$ and $TX$ is the tangent bundle of $X$
• Chern numbers are important in the classification of complex manifolds and the study of their topological and geometric properties

Chern classes and Riemann-Roch theorem

• The Riemann-Roch theorem is a fundamental result in algebraic geometry that relates the Euler characteristic of a coherent sheaf on a complex algebraic variety to the Chern classes of the sheaf and the variety
• For a compact complex manifold $X$ and a coherent sheaf $\mathcal{F}$ on $X$, the Riemann-Roch theorem states that $\chi(\mathcal{F}) = \int_X ch(\mathcal{F}) \cdot td(TX)$, where $\chi(\mathcal{F})$ is the Euler characteristic of $\mathcal{F}$, $ch(\mathcal{F})$ is the Chern character of $\mathcal{F}$, and $td(TX)$ is the Todd class of the tangent bundle of $X$
• The Riemann-Roch theorem has numerous applications in the study of algebraic curves, surfaces, and higher-dimensional varieties

Generalized Chern classes

• The concept of Chern classes has been generalized and extended in various directions to study more general settings and objects
• These generalizations provide powerful tools for investigating the geometry and topology of manifolds, bundles, and other mathematical structures

Chern-Simons classes

• Chern-Simons classes are secondary characteristic classes associated with principal bundles and their connections
• They are defined in odd-dimensional cohomology groups and are related to the Chern classes via the Chern-Simons transgression formula
• Chern-Simons classes play a crucial role in the study of three-dimensional manifolds, knot theory, and mathematical physics

Chern-Weil theory

• Chern-Weil theory is a general framework for constructing characteristic classes of principal bundles using connections and curvature
• It provides a unified approach to defining and studying various characteristic classes, including Chern classes, Pontryagin classes, and Euler classes
• The Chern-Weil homomorphism relates the invariant polynomials on the Lie algebra of the structure group to the cohomology of the base space

Equivariant Chern classes

• Equivariant Chern classes are a generalization of Chern classes to the setting of equivariant vector bundles, which are vector bundles with a compatible group action
• They are elements of the equivariant cohomology of the base space, which takes into account the group action
• Equivariant Chern classes have applications in the study of group actions on manifolds and in equivariant K-theory

Chern classes in K-theory

• K-theory is a generalized cohomology theory that assigns abelian groups to topological spaces, with vector bundles as the primary objects of study
• Chern classes can be defined in K-theory, where they provide a connection between the K-theory of a space and its ordinary cohomology
• The Chern character is a natural transformation from K-theory to ordinary cohomology that relates the Chern classes in K-theory to the classical Chern classes

Chern classes in algebraic geometry

• In algebraic geometry, Chern classes are defined for algebraic vector bundles over algebraic varieties
• They are elements of the Chow ring, which is an algebraic analogue of the cohomology ring
• Chern classes in algebraic geometry have applications in the study of algebraic cycles, intersection theory, and the geometry of algebraic varieties
• The Grothendieck-Riemann-Roch theorem is a generalization of the Riemann-Roch theorem that relates Chern classes in algebraic geometry to the K-theory of coherent sheaves