Chern Character

Definition

The Chern character is a topological invariant associated with a complex vector bundle, providing a way to relate the geometry of the bundle to its cohomology. It serves as a bridge between Chern classes and characteristic classes, enabling the extraction of significant topological information from complex manifolds. The Chern character is particularly important in understanding the Riemann-Roch theorem and its applications in algebraic geometry and topology.

5 Must Know Facts For Your Next Test

1. The Chern character is computed as the trace of the exponential of the Chern class, which captures the topological features of the complex vector bundle.
2. It is an additive function on vector bundles, meaning that for two bundles, the Chern character of their direct sum equals the sum of their individual Chern characters.
3. In terms of cohomology, the Chern character provides a homomorphism from the Grothendieck group of vector bundles to the rational cohomology ring.
4. The Chern character can be expressed in terms of Chern classes via the formula: $$ ext{ch}(E) = ext{e}^{c_1(E)} + rac{1}{2} ext{e}^{c_2(E)} + ...$$ where $c_i$ are the Chern classes.
5. The Chern character plays a key role in computing integrals over complex manifolds, especially in applications related to physics and string theory.

Review Questions

• How does the Chern character relate to Chern classes and why is this relationship significant?
  • The Chern character is defined using Chern classes, specifically as a function that summarizes their contributions to the topology of complex vector bundles. This relationship is significant because it allows us to utilize Chern classes' geometric insights while applying the more algebraic properties captured by the Chern character. Essentially, the Chern character gives a deeper understanding of how these classes reflect on cohomological properties and contribute to various results in topology.
• Describe how the Chern character can be used in practical applications within algebraic geometry or topology.
  • In algebraic geometry, the Chern character is essential for applying the Riemann-Roch theorem, which connects geometric properties of varieties to their cohomological features. It helps in computing dimensions of space sections and allows researchers to understand how topological invariants can influence algebraic structures. This application extends into areas like string theory and mathematical physics, where understanding complex manifolds' geometry is crucial for model building and theoretical development.
• Evaluate the implications of the Chern character being an additive function on vector bundles. What does this mean for their classification?
  • The additivity property of the Chern character implies that when we take direct sums of vector bundles, their topological features can be effectively combined. This has major implications for classifying vector bundles since it means we can study simpler or more fundamental bundles and build up more complex structures from them. Consequently, this property aids in constructing invariants that help distinguish different types of bundles and provides tools for proving results related to intersection theory and characteristic classes.

Related terms

Chern Classes: Chern classes are characteristic classes associated with complex vector bundles, providing a way to measure the curvature of the bundle and its topological properties.

Characteristic Class: Characteristic classes are a set of invariants that classify vector bundles and provide a way to understand their topology in terms of cohomology.

Riemann-Roch Theorem: The Riemann-Roch theorem relates the geometry of a complex manifold to its cohomological properties, allowing for the calculation of dimensions of spaces of sections of line bundles.