Equivariant D-Modules and Local Cohomology in Algebraic Geometry

Equivariant D-modules combine the algebraic theory of differential operators with group actions on algebraic varieties, providing a framework to study systems of linear partial differential equations that respect symmetries. In parallel, local cohomology probes the structure of sheaves in neighbourhoods of closed subsets, capturing support conditions and depth properties crucial in singularity theory. The intersection of these domains yields powerful tools for representation theory, Hodge theory and mirror symmetry: equivariant structures constrain the behaviour of D-module solutions under group or torus actions, while local cohomology modules extract finiteness, duality and vanishing phenomena. Recent advances exploit derived and microlocal methods to classify holonomic equivariant modules on flag varieties, relate their characteristic cycles to equivariant intersection cohomology, and compute local cohomology in graded settings such as toric and multigraded rings. The synergy of these approaches illuminates duality theorems, Riemann–Hilbert correspondences in the equivariant context and p-adic analogues, with applications ranging from counting rational points on varieties to the study of special functions in several variables.

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Technical terms

D-Module: A sheaf of modules over the sheaf of differential operators on an algebraic variety, encoding systems of linear PDEs.

Holonomic D-Module: A D-module whose characteristic variety has minimal possible dimension, ensuring finiteness of solution spaces and duality properties.

Equivariant D-Module: A D-module equipped with a compatible action of an algebraic group, preserving differential structure under symmetry transformations.

Local Cohomology: Cohomology theory focusing on sections supported in a specified closed subset, capturing depth and local duality of sheaves.

References

1. Differential operators, retracts, and toric face rings. Algebra & Number Theory (2023).
2. Inductive system coherence for logarithmic arithmetic ${\mathcal D}$-modules, stability for cohomology operations. Documenta Mathematica (2016).
3. Regularization of Relative Holonomic $\mathcal{D}$-Modules. Publications of the Research Institute for Mathematical Sciences (2024).