Chow Group

Definition

Chow groups are mathematical structures that capture the idea of algebraic cycles on a projective variety, allowing for the classification and study of these cycles in a rigorous way. They provide a way to connect geometry with algebraic topology and help in understanding properties of varieties, especially in the context of intersection theory and motives.

5 Must Know Facts For Your Next Test

1. Chow groups are typically denoted by $CH^i(X)$ for an algebraic variety $X$, where $i$ indicates the dimension of the cycles being considered.
2. The Chow group is equipped with a natural structure of abelian groups, allowing for operations such as addition and scalar multiplication.
3. There is a correspondence between algebraic cycles and cohomology classes, making Chow groups a bridge between algebraic geometry and topology.
4. Chow groups can be used to define intersection numbers, providing important invariants of varieties that reveal geometric information.
5. The study of Chow groups leads to deeper insights in the field of motives, which aim to unify various cohomology theories and provide a framework for understanding the relationships between them.

Review Questions

• How do Chow groups relate to algebraic cycles and what role do they play in understanding the properties of projective varieties?
  • Chow groups serve as a formal framework for analyzing algebraic cycles on projective varieties by providing a way to classify these cycles mathematically. They capture important properties such as intersection numbers and help in exploring relationships between different cycles. This relationship enhances our understanding of the geometry of varieties, particularly in how different subvarieties intersect and interact within the larger structure.
• Discuss the significance of motivic cohomology in relation to Chow groups and how it contributes to the study of algebraic geometry.
  • Motivic cohomology plays a crucial role alongside Chow groups by offering a broader context in which these cycles can be studied. It provides tools from both algebraic geometry and topology, allowing for the comparison of various cohomological invariants. This interplay helps to uncover deeper connections between geometry and arithmetic properties of varieties, as well as facilitating new insights into classical problems in algebraic geometry.
• Evaluate the implications of intersection theory on Chow groups and how this connection enhances our understanding of algebraic varieties.
  • Intersection theory significantly impacts the study of Chow groups by allowing us to calculate intersection numbers, which are essential invariants associated with algebraic cycles. This connection provides valuable geometric information about how subvarieties intersect within a larger variety. By analyzing these intersections through the lens of Chow groups, mathematicians gain insights into the structure and behavior of algebraic varieties, which is fundamental for advancing theories in both algebraic geometry and arithmetic.

Related terms

Algebraic Cycle: An algebraic cycle is a formal sum of subvarieties of an algebraic variety, where coefficients are integers, representing geometric objects that can be studied in relation to Chow groups.

Motivic Cohomology: Motivic cohomology is a type of cohomology theory that captures information about algebraic varieties using tools from algebraic geometry and homotopy theory, closely related to Chow groups.

Intersection Theory: Intersection theory studies how subvarieties intersect within a given variety, playing a crucial role in defining Chow groups and understanding their properties.