Pushforward

Definition

Pushforward refers to a mathematical operation that takes a cycle (a formal sum of subvarieties) from one space and maps it to another space via a continuous function or morphism. This operation is significant in understanding how geometric objects transform under various mappings and is especially important in the context of cycle class maps, where it helps describe how the algebraic cycles behave with respect to changes in their ambient spaces.

5 Must Know Facts For Your Next Test

1. The pushforward operation is typically denoted by the symbol $f_*$, where $f$ is the morphism mapping the cycles between spaces.
2. In the context of algebraic geometry, pushforwards help in defining and computing intersection numbers, as they enable the translation of cycles across different varieties.
3. Pushforward can also be understood in terms of measures when considering cycles as measures on the target space, providing a bridge between geometry and analysis.
4. This operation respects certain properties such as additivity and compatibility with proper maps, ensuring that it behaves well under various geometric scenarios.
5. Pushforwards play a crucial role in the formulation of Chern classes and other characteristic classes, linking topology with algebraic geometry through these mappings.

Review Questions

• How does the pushforward operation affect cycles when applied through a morphism, and what implications does this have for their geometric properties?
  • When the pushforward operation is applied to cycles via a morphism, it translates the geometric properties of those cycles into the target space. This means that characteristics such as dimension and intersection behavior may change depending on how the cycle interacts with the mapping. Essentially, it allows us to understand how cycles 'move' in a geometrical sense under transformations, thus providing insight into their behavior across different spaces.
• Discuss how pushforward interacts with pullback and why both operations are essential in studying algebraic cycles.
  • Pushforward and pullback are dual operations that serve complementary roles in studying algebraic cycles. While pushforward moves cycles from one space to another, pullback brings co-cycles back from the target space. This interaction is crucial because it allows for the computation of invariants and properties of cycles across different varieties, facilitating a deeper understanding of their relationships. The interplay between these operations reveals insights about how geometric structures are preserved or altered under mappings.
• Evaluate the importance of pushforward in connecting algebraic geometry with topology, particularly in defining Chern classes.
  • The importance of pushforward lies in its ability to bridge algebraic geometry with topology through operations like defining Chern classes. These classes encapsulate topological features of vector bundles over varieties and rely heavily on the pushforward operation to translate these features into an algebraic framework. By mapping cycles from one space to another, pushforward helps articulate how geometric properties manifest topologically, thus enriching our understanding of both disciplines and allowing for robust applications in theoretical contexts.

Related terms

Cycle: A cycle is a formal sum of subvarieties (or algebraic subspaces) within a variety, which can be used to study the geometry and topology of the variety.

Pullback: Pullback is an operation that takes a co-cycle from one space and 'pulls it back' to another space via a continuous function or morphism, often used alongside pushforward.

Morphism: A morphism is a structure-preserving map between two mathematical objects, such as varieties, that allows for the transfer of information and properties between them.