Rational Singularities

Definition

Rational singularities are a type of singularity in algebraic geometry where the local cohomology of a space is 'rational' in the sense that it can be computed using a rational function. This concept is particularly important because it provides insights into the geometry of complex algebraic varieties and their behavior under various transformations, linking ideas from both algebraic geometry and symplectic geometry.

5 Must Know Facts For Your Next Test

1. Rational singularities occur when the local cohomology groups at the singular point are finitely generated and exhibit rationality, which means they can be expressed in terms of rational numbers.
2. These singularities are important in the context of resolutions, as they often admit resolutions that lead to 'nice' geometric structures.
3. In the study of rational singularities, one key criterion is whether the variety can be deformed into a smooth variety, revealing insights about its underlying structure.
4. Rational singularities have implications for Hodge theory, particularly in understanding the behavior of differential forms near singular points.
5. The connection between rational singularities and symplectic geometry arises from their influence on the deformation theory of complex structures, linking algebraic and geometric aspects.

Review Questions

• How do rational singularities relate to the concept of local cohomology in algebraic geometry?
  • Rational singularities are closely tied to local cohomology because they define specific conditions under which the local cohomology groups at singular points are finitely generated and rational. This property allows for a deeper understanding of the structure and behavior of algebraic varieties near these points. The relationship indicates that if a variety has rational singularities, it possesses a certain level of 'goodness' that facilitates further analysis and potential resolution.
• Discuss the importance of resolving rational singularities in relation to smooth varieties and how this affects their geometric properties.
  • Resolving rational singularities is crucial because it allows algebraic varieties with such singularities to be transformed into smooth varieties, which possess better geometric properties. The resolution process often reveals how these singularities can be tamed or managed, providing insights into how the original variety behaves under deformation. This transformation aids in understanding the overall structure and ensures that various geometric properties, such as Hodge structures, remain intact after resolution.
• Evaluate the role of rational singularities in linking symplectic geometry with complex algebraic varieties, particularly concerning deformation theory.
  • Rational singularities play a significant role in connecting symplectic geometry with complex algebraic varieties through deformation theory. They help in understanding how complex structures on varieties can change while preserving essential geometric properties. By studying rational singularities, mathematicians can uncover deeper relationships between algebraic properties and symplectic forms, allowing for advancements in both fields and showing how the deformation theory helps reconcile complex analytical techniques with geometric intuitions.

Related terms

Cohomology:

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

Singularities: Points on a mathematical object where it fails to be well-behaved, such as points where a function is not differentiable or where a space does not look locally like Euclidean space.

Complex Algebraic Varieties: Geometric objects defined as the solutions to systems of polynomial equations in complex numbers, serving as fundamental objects of study in algebraic geometry.