Canonical divisor

Definition

A canonical divisor is a divisor associated with a variety that encodes important geometric and topological information about the variety itself. It is often denoted by $K_X$ and can be viewed as a generalization of the notion of canonical forms in algebraic geometry, reflecting the duality between divisors and differentials. Understanding canonical divisors plays a crucial role in analyzing singularities, vanishing theorems, and applying the Riemann-Roch theorem to curves and surfaces.

5 Must Know Facts For Your Next Test

1. The canonical divisor $K_X$ is defined as the divisor associated with the canonical line bundle, which is generated by differentials of top degree on the variety.
2. In the context of singularities, a variety with terminal or canonical singularities has its canonical divisor behaving in specific ways that influence its structure and classification.
3. Kodaira's vanishing theorem involves the properties of the canonical divisor, asserting that certain cohomology groups vanish under specific conditions, leading to implications for algebraic cycles.
4. The Riemann-Roch theorem for curves and surfaces uses canonical divisors to derive formulas for computing dimensions of spaces of sections, highlighting their importance in intersection theory.
5. The study of canonical divisors is crucial in determining birational equivalence classes of varieties and understanding their geometric properties.

Review Questions

• How does the concept of a canonical divisor relate to singularities, and what implications does this have for classifying these singularities?
  • The canonical divisor plays a key role in determining the type of singularities present on a variety. Specifically, when analyzing terminal and canonical singularities, the behavior of the canonical divisor helps classify these singularities based on how they affect the geometry and smoothness of the variety. If a variety has a well-behaved canonical divisor, it suggests that it has more regular geometric properties, leading to important implications in birational geometry.
• Discuss how Kodaira vanishing theorem utilizes the properties of canonical divisors to draw conclusions about cohomology groups on varieties.
  • Kodaira vanishing theorem leverages the structure of the canonical divisor by asserting that if certain conditions are met—like having ample line bundles—the higher cohomology groups vanish. This connects to how sections behave under tensor products with the canonical line bundle. The results from this theorem provide significant insights into geometric properties and aid in computations regarding vector bundles over algebraic varieties.
• Evaluate the importance of canonical divisors in formulating the Riemann-Roch theorem for both curves and surfaces.
  • Canonical divisors are central to formulating the Riemann-Roch theorem as they directly influence calculations regarding dimension and sections. For curves, the relationship between effective divisors and the canonical divisor leads to a powerful formula that can predict dimensions based on genus. In surfaces, similar principles apply, where understanding how these divisors interact can yield critical results about intersection numbers and geometry. Thus, they are vital tools for connecting algebraic properties to geometric intuition in both contexts.

Related terms

Divisor:

A divisor is a formal sum of codimension-one subvarieties of a given variety, which helps in studying the function theory on varieties.

Riemann-Roch Theorem:

A fundamental result in algebraic geometry that relates the dimensions of spaces of sections of line bundles to the geometry of the underlying variety.

Kodaira Dimension:

The Kodaira dimension is an invariant that classifies complex projective varieties based on the growth of their sections of powers of the canonical line bundle.