Goresky-MacPherson Formula

Definition

The Goresky-MacPherson formula is a powerful result in algebraic geometry that relates the intersection cohomology of a singular variety to its topological properties. It provides a way to compute the intersection cohomology of complex algebraic varieties, particularly those that are not smooth, using their stratifications. This formula is especially relevant when dealing with toric varieties, as it allows for the calculation of their cohomology using combinatorial data from the associated fan.

5 Must Know Facts For Your Next Test

1. The Goresky-MacPherson formula specifically computes the intersection cohomology groups by incorporating information about the stratification of the space.
2. This formula generalizes earlier results on smooth varieties, providing tools to handle singularities in algebraic geometry.
3. In the context of toric varieties, the Goresky-MacPherson formula allows for explicit calculations of their intersection cohomology using the fan associated with the variety.
4. The formula relies on various mathematical tools such as sheaf theory and Poincaré duality to establish connections between topology and algebraic geometry.
5. Applications of the Goresky-MacPherson formula extend beyond algebraic geometry into areas like mathematical physics and mirror symmetry.

Review Questions

• How does the Goresky-MacPherson formula enhance our understanding of singular varieties in algebraic geometry?
  • The Goresky-MacPherson formula enhances our understanding of singular varieties by providing a method to compute their intersection cohomology, which captures important topological information. This is particularly useful because traditional methods often fail for singular spaces. By employing stratifications and considering various dimensions within these varieties, the formula allows mathematicians to analyze and derive properties that would otherwise be obscured by singularities.
• Discuss how the application of the Goresky-MacPherson formula to toric varieties differs from its application to general singular varieties.
  • When applying the Goresky-MacPherson formula to toric varieties, one can leverage their combinatorial structure defined by fans, which simplifies calculations significantly. In contrast, for general singular varieties, more complex stratifications and additional topological considerations may be needed. The toric case provides an explicit framework where one can directly relate the combinatorial data from the fan to the intersection cohomology groups, highlighting the power of this formula in specific contexts.
• Evaluate the significance of the Goresky-MacPherson formula in bridging algebraic geometry with other areas of mathematics.
  • The significance of the Goresky-MacPherson formula lies in its ability to bridge algebraic geometry with topology and mathematical physics, showcasing its interdisciplinary impact. By providing a coherent framework for understanding intersection cohomology, it opens up pathways for applications in mirror symmetry and string theory. The connections made through this formula enable mathematicians to utilize geometric insights across different mathematical domains, fostering collaboration and new discoveries at their intersections.

Related terms

Intersection Cohomology: A generalized cohomology theory that extends ordinary cohomology to include singular spaces, allowing for a better understanding of their topological features.

Toric Varieties: Algebraic varieties that are constructed from combinatorial data associated with fans, which play a crucial role in the study of algebraic geometry and mirror symmetry.

Stratification: A decomposition of a space into disjoint subsets called strata, where each stratum has a well-defined geometric structure, often used to study singular spaces.