Chern character

Definition

The Chern character is a topological invariant associated with complex vector bundles, providing a way to study their geometry through cohomology. It captures important information about the curvature and topology of the bundles, and it is closely related to other invariants such as Chern classes. By connecting the geometric properties of fiber bundles and differentiable structures, the Chern character plays a significant role in understanding various aspects of topology and geometry.

5 Must Know Facts For Your Next Test

1. The Chern character is defined as the exponential of the Chern classes, specifically expressed as a polynomial in these classes.
2. It is particularly useful in computing intersection numbers on complex manifolds and has applications in enumerative geometry.
3. The Chern character is a homomorphism from the Grothendieck group of complex vector bundles to cohomology rings, preserving important algebraic structures.
4. For complex line bundles, the Chern character can be computed using the first Chern class, showing its foundational role in understanding more complex bundles.
5. The integration of the Chern character over a manifold can yield significant geometric information, linking topology with analytical properties.

Review Questions

• How does the Chern character relate to Chern classes and what significance does this relationship have in topology?
  • The Chern character is directly related to Chern classes as it is essentially defined using these classes; specifically, it involves taking the exponential of the total Chern class. This relationship allows us to translate geometric data captured by Chern classes into topological invariants through cohomology. The significance lies in how this connection helps us understand complex vector bundles and their curvature, providing deeper insights into both algebraic and differential topology.
• Discuss how the Chern character can be utilized in computations related to complex manifolds and their topology.
  • The Chern character plays a crucial role in computations involving intersection numbers on complex manifolds. By utilizing its relationship with Chern classes, one can derive results about how different curves intersect on these manifolds. This becomes especially valuable in enumerative geometry, where one seeks to count specific geometric configurations, making the Chern character an essential tool for linking topology with practical calculations in geometry.
• Evaluate the impact of the Chern character on understanding the geometry of fiber bundles and its implications for modern mathematical research.
  • The impact of the Chern character on understanding fiber bundles is profound as it provides a bridge between differential geometry and algebraic topology. By offering insights into how curvature affects global properties of bundles, it enhances our comprehension of various geometric structures. Modern mathematical research benefits from this as it leads to advancements in areas like gauge theory and string theory, where such topological invariants are pivotal for formulating theories that describe fundamental forces in physics.

Related terms

Chern classes: Chern classes are characteristic classes that provide a way to measure the curvature of complex vector bundles, reflecting their topological properties.

Fiber bundle: A fiber bundle is a structure that consists of a base space, a total space, and a typical fiber, allowing for local trivialization while encoding global topological information.

Stiefel-Whitney classes: Stiefel-Whitney classes are characteristic classes associated with real vector bundles, serving as invariants in the study of manifold topology.