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Local cohomology with support in Schubert varieties

Michael Perlman

TL;DR

The paper develops a Hodge-theoretic framework for local cohomology with support in Schubert varieties on generalized flag varieties, recasting the problem through the Grothendieck–Cousin complex and Beilinson–Bernstein localization to parabolic Verma modules. By upgrading the GC complex to a complex of mixed Hodge modules, the authors obtain degeneration of the Hodge and weight filtrations and thereby compute composition factors and filtrations explicitly. In the Grassmannian case, they translate these filtrations into combinatorial data of augmented Dyck patterns, linking weight structures to Raicu–Weyman’s patterns and, via super duality, to syzygies of determinantal thickenings. They further apply the results to determinantal varieties by restricting to the opposite big cell, recovering known composition factors and weight filtrations in a new, combinatorially transparent form. Overall, the work provides a robust bridge between local cohomology, D-modules, Hodge theory, and representation-theoretic/combinatorial tools, yielding explicit descriptions of filtrations and composition factors with broad applications to determinantal geometry.

Abstract

This paper is concerned with local cohomology sheaves on generalized flag varieties supported in closed Schubert varieties, which carry natural structures as (mixed Hodge) D-modules. We employ Kazhdan--Lusztig theory and Saito's theory of mixed Hodge modules to describe a general strategy to calculate the simple composition factors, Hodge filtration, and weight filtration on these modules. Our main tool is the Grothendieck--Cousin complex, introduced by Kempf, which allows us to relate the local cohomology modules in question to parabolic Verma modules over the corresponding Lie algebra. We show that this complex underlies a complex of mixed Hodge modules, and is thus endowed with Hodge and weight filtrations. As a consequence, strictness implies that computing cohomology commutes with taking associated graded with respect to both of these filtrations. We execute this strategy to calculate the composition factors and weight filtration for Schubert varieties in the Grassmannian, in particular showing that the weight filtration is controlled by the augmented Dyck patterns of Raicu--Weyman. As an application, upon restriction to the opposite big cell, we recover the simple composition factors and weight filtration on local cohomology with support in generic determinantal varieties.

Local cohomology with support in Schubert varieties

TL;DR

The paper develops a Hodge-theoretic framework for local cohomology with support in Schubert varieties on generalized flag varieties, recasting the problem through the Grothendieck–Cousin complex and Beilinson–Bernstein localization to parabolic Verma modules. By upgrading the GC complex to a complex of mixed Hodge modules, the authors obtain degeneration of the Hodge and weight filtrations and thereby compute composition factors and filtrations explicitly. In the Grassmannian case, they translate these filtrations into combinatorial data of augmented Dyck patterns, linking weight structures to Raicu–Weyman’s patterns and, via super duality, to syzygies of determinantal thickenings. They further apply the results to determinantal varieties by restricting to the opposite big cell, recovering known composition factors and weight filtrations in a new, combinatorially transparent form. Overall, the work provides a robust bridge between local cohomology, D-modules, Hodge theory, and representation-theoretic/combinatorial tools, yielding explicit descriptions of filtrations and composition factors with broad applications to determinantal geometry.

Abstract

This paper is concerned with local cohomology sheaves on generalized flag varieties supported in closed Schubert varieties, which carry natural structures as (mixed Hodge) D-modules. We employ Kazhdan--Lusztig theory and Saito's theory of mixed Hodge modules to describe a general strategy to calculate the simple composition factors, Hodge filtration, and weight filtration on these modules. Our main tool is the Grothendieck--Cousin complex, introduced by Kempf, which allows us to relate the local cohomology modules in question to parabolic Verma modules over the corresponding Lie algebra. We show that this complex underlies a complex of mixed Hodge modules, and is thus endowed with Hodge and weight filtrations. As a consequence, strictness implies that computing cohomology commutes with taking associated graded with respect to both of these filtrations. We execute this strategy to calculate the composition factors and weight filtration for Schubert varieties in the Grassmannian, in particular showing that the weight filtration is controlled by the augmented Dyck patterns of Raicu--Weyman. As an application, upon restriction to the opposite big cell, we recover the simple composition factors and weight filtration on local cohomology with support in generic determinantal varieties.
Paper Structure (17 sections, 19 theorems, 102 equations, 30 figures)

This paper contains 17 sections, 19 theorems, 102 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Statement of results for the Grassmannian
  4. The Hodge-theoretic Grothendieck-Cousin complex
  5. $\mathcal{D}$-modules, mixed Hodge modules, and local cohomology
  6. Construction of the Grothendieck--Cousin complex
  7. The Grothendieck--Cousin complex of a Schubert variety
  8. Lie algebras and Weyl groups
  9. Generalized flag varieties and Schubert varieties
  10. Structure of the Grothendieck--Cousin complex
  11. Parabolic Verma modules, $P$-orbits, and localization
  12. Calculations for the Grassmannian
  13. The Grassmannian and minimal length coset representatives
  14. Properties of the parabolic Verma modules
  15. Cohomology of the Grothendieck--Cousin complex
  16. ...and 2 more sections

Key Result

Theorem 1.1

With the notation above, we have the following expression in the Grothendieck group of $\mathcal{D}_{X}$-modules (resp. representations of $\mathfrak{sl}_{n}(\mathbb{C}))$ for $q\geq 0$: Furthermore, the following is true about the weight filtration $W_{\bullet}$ for $p\in \mathbb{Z}$:

Figures (30)

  • Figure 1: $(5,4,2,2)$
  • Figure 2: $(3,3,2,2)$
  • Figure 3: $(3,1,1,1)$
  • Figure 4: $(5,1,1,1)$
  • Figure 5: $(0,0,0,0)$
  • ...and 25 more figures

Theorems & Definitions (42)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • ...and 32 more