Perverse Sheaf
Introduction
In the realm of algebraic geometry and topology, the concept of a perverse sheaf emerges as a sophisticated tool that bridges the gap between sheaf theory and intersection homology. Perverse sheaves provide a framework for understanding the topology of singular spaces, offering a refined perspective on the derived category of sheaves. This article delves into the intricate nature of perverse sheaves, exploring their definitions, properties, and applications within the broader context of modern mathematics.
Historical Context
The notion of perverse sheaves was introduced in the 1980s by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne as part of their work on the Weil conjectures. Their development was motivated by the need to extend the Riemann-Hilbert correspondence to singular spaces. The introduction of perverse sheaves marked a significant advancement in the study of singularities and stratified spaces, providing a robust tool for analyzing cohomology theories.
Definition and Basic Properties
A perverse sheaf is a complex of sheaves that satisfies certain cohomological conditions relative to a given stratification of a space. More formally, let \( X \) be a complex algebraic variety or a topological space with a fixed stratification. A perverse sheaf on \( X \) is an object in the derived category of sheaves on \( X \) that satisfies specific conditions on its cohomology sheaves with respect to the stratification.
These conditions are expressed in terms of the t-structure on the derived category, which is a pair of full subcategories that generalize the notion of cohomological degree. The perverse t-structure is defined such that the perverse sheaves are the heart of this t-structure. The conditions involve the support and cosupport of the cohomology sheaves, ensuring that they are concentrated in certain degrees relative to the dimension of the strata.
The Perverse t-Structure
The perverse t-structure is a central concept in the theory of perverse sheaves. It is defined on the derived category of sheaves \( D^b(X) \) and consists of two subcategories: the aisle \( {}^pD^{\leq 0}(X) \) and the coaisle \( {}^pD^{\geq 0}(X) \). These subcategories are defined by the following conditions:
- A complex \( \mathcal{F} \in D^b(X) \) belongs to \( {}^pD^{\leq 0}(X) \) if for every stratum \( S \) of \( X \), the cohomology sheaves \( \mathcal{H}^i(\mathcal{F}) \) satisfy \( \text{dim supp} \mathcal{H}^i(\mathcal{F}) \leq -i \). - A complex \( \mathcal{F} \in D^b(X) \) belongs to \( {}^pD^{\geq 0}(X) \) if for every stratum \( S \) of \( X \), the cohomology sheaves \( \mathcal{H}^i(\mathcal{F}) \) satisfy \( \text{dim cosupp} \mathcal{H}^i(\mathcal{F}) \geq -i \).
The heart of the perverse t-structure, denoted \( \text{Perv}(X) \), consists of those complexes that are simultaneously in \( {}^pD^{\leq 0}(X) \) and \( {}^pD^{\geq 0}(X) \).
Examples of Perverse Sheaves
Perverse sheaves can be found in various mathematical contexts, often arising naturally in the study of singular spaces. Some notable examples include:
- **Intersection Cohomology Complexes:** These are perverse sheaves that generalize the notion of intersection homology, providing a way to compute the intersection cohomology of singular spaces. - **Local Systems on Smooth Strata:** If \( X \) is a smooth manifold, a local system on \( X \) is a perverse sheaf. More generally, if \( X \) is stratified, a local system on each stratum that satisfies certain vanishing conditions can be extended to a perverse sheaf on \( X \). - **Constant Sheaf on a Point:** The constant sheaf on a point is trivially a perverse sheaf, serving as a simple example to illustrate the basic properties of perverse sheaves.
Applications in Algebraic Geometry
Perverse sheaves play a pivotal role in algebraic geometry, particularly in the study of singular varieties and moduli spaces. They provide a powerful tool for understanding the topology of these spaces, allowing for the computation of intersection cohomology and the analysis of monodromy representations.
One of the key applications of perverse sheaves is in the proof of the Decomposition Theorem, which asserts that the direct image of a perverse sheaf under a proper map decomposes into a direct sum of shifted perverse sheaves. This theorem has profound implications for the study of Hodge theory and the topology of algebraic varieties.
Perverse Sheaves and D-Modules
The relationship between perverse sheaves and D-modules is a central theme in the study of the Riemann-Hilbert correspondence. D-modules provide an algebraic framework for understanding differential equations on algebraic varieties, and perverse sheaves offer a topological perspective on these equations.
The Riemann-Hilbert correspondence establishes an equivalence between the category of regular holonomic D-modules and the category of perverse sheaves. This correspondence allows for the translation of problems in algebraic analysis into the language of topology, providing new insights into the structure of solutions to differential equations.
Perverse Sheaves in Representation Theory
In representation theory, perverse sheaves are used to study the geometry of flag varieties and Schubert varieties. The Kazhdan-Lusztig conjectures, which relate the representation theory of Hecke algebras to the geometry of these varieties, are formulated in terms of perverse sheaves.
The Springer correspondence, which connects representations of the symmetric group to the geometry of the nilpotent cone, is another example of the interplay between perverse sheaves and representation theory. Perverse sheaves provide a geometric framework for understanding the structure of these representations, revealing deep connections between algebra and geometry.
Challenges and Open Questions
Despite their success in various areas of mathematics, the theory of perverse sheaves continues to present challenges and open questions. One of the main difficulties lies in the explicit computation of perverse sheaves on complex spaces, particularly in higher dimensions.
The development of computational tools and algorithms for perverse sheaves remains an active area of research, with potential applications in mirror symmetry and string theory. Additionally, the extension of perverse sheaf theory to non-archimedean and p-adic geometry is an ongoing area of exploration, with potential implications for number theory and arithmetic geometry.