Intersection form

Definition

The intersection form is a bilinear form associated with the intersection of cycles in algebraic geometry, specifically in the context of a smooth projective variety. It provides a way to compute how subvarieties intersect with one another and encapsulates important topological information about the variety. Understanding the intersection form is crucial for studying the geometry of cycles and their classes in cohomology, which are key concepts in both algebraic and differential geometry.

5 Must Know Facts For Your Next Test

1. The intersection form can be represented as a matrix whose entries indicate how different cycles intersect within the variety.
2. In a smooth projective variety, the intersection form is often non-degenerate, meaning it provides rich geometric information.
3. The properties of the intersection form are closely tied to the topology of the underlying space, particularly through Poincaré duality.
4. For complete intersections, the intersection form can be computed using intersection theory, which generalizes classical results from algebraic geometry.
5. The intersection form has applications in various fields, including mirror symmetry and string theory, highlighting its significance beyond pure mathematics.

Review Questions

• How does the intersection form provide insights into the relationships between different cycles within a smooth projective variety?
  • The intersection form acts as a bilinear mapping that reveals how different cycles intersect with one another within a smooth projective variety. By examining the matrix representation of the intersection form, one can determine the number of points where cycles meet, as well as their respective dimensions and orientations. This information is essential for understanding the geometric structure of the variety and helps characterize its cohomological properties.
• Discuss the significance of non-degeneracy in the intersection form and its implications for smooth projective varieties.
  • Non-degeneracy in the intersection form means that it gives a perfect pairing between cycles, allowing for meaningful computations regarding intersections. In smooth projective varieties, this property ensures that each cycle can be represented uniquely by its cohomology class, enabling powerful tools like Poincaré duality to apply. This uniqueness is crucial for understanding how cycles behave under various geometric operations and for establishing deeper connections between geometry and topology.
• Evaluate how the intersection form contributes to broader areas of research such as mirror symmetry and string theory.
  • The intersection form plays a pivotal role in areas like mirror symmetry and string theory by providing a framework for analyzing complex geometric structures. In mirror symmetry, it helps relate two seemingly different geometrical contexts by demonstrating how their cycles interact through intersections. In string theory, understanding these intersections aids in the compactification of extra dimensions, leading to significant implications in theoretical physics. This interplay between geometry and physical theories illustrates the importance of the intersection form beyond classical mathematics.

Related terms

Cohomology:

A mathematical tool used to study topological spaces through algebraic invariants, providing insights into the structure and properties of spaces.

Cycle: A formal sum of subvarieties or submanifolds in a given space, which is used to define intersections and perform geometric operations.

Bilinear Form: A function that is linear in each of its arguments separately, often used to study the relationships between vector spaces or other algebraic structures.