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A decomposition theorem for Lefschetz modules

Omid Amini, June Huh, Matt Larson

TL;DR

The paper develops an algebraic decomposition framework for Lefschetz modules over graded algebras, capturing Poincaré duality, Hard Lefschetz, and Hodge--Riemann phenomena in a purely algebraic setting. It introduces a decomposition package that fixes an indecomposable Krull–Schmidt structure and a perverse filtration whose associated graded $\operatorname{Gr}$ inherits rich Lefschetz and Hodge-type properties, including relative Hard Lefschetz and relative Hodge--Riemann theorems. The results are then applied across combinatorics (matroids and polytopes) and geometry (projective varieties, Chow rings, and intersection cohomology), yielding new proofs and extensions of known Lefschetz phenomena and enabling polarization results for intersection cohomology via the decomposition framework. Overall, the work provides a unifying algebraic approach to the decomposition theorem’s Hodge-theoretic aspects, with implications for standard conjectures and polarized Hodge structures in geometric contexts.

Abstract

A Lefschetz module is a module over a graded algebra $A$ that satisfies analogues of Poincaré duality, the Hard Lefschetz property, and the Hodge--Riemann relations with respect to an open convex cone $\mathscr{K}$ in the degree one part of $A$. We analyze its decomposition into indecomposable modules over subrings of $A$ that are generated by elements in the closure of $\mathscr{K}$, establishing structural results that parallel the decomposition theorem for morphisms of complex projective varieties. We use our theorems to recover key statements in combinatorial Hodge theory and illuminate the Hodge-theoretic aspects of the decomposition theorem in algebraic geometry.

A decomposition theorem for Lefschetz modules

TL;DR

The paper develops an algebraic decomposition framework for Lefschetz modules over graded algebras, capturing Poincaré duality, Hard Lefschetz, and Hodge--Riemann phenomena in a purely algebraic setting. It introduces a decomposition package that fixes an indecomposable Krull–Schmidt structure and a perverse filtration whose associated graded inherits rich Lefschetz and Hodge-type properties, including relative Hard Lefschetz and relative Hodge--Riemann theorems. The results are then applied across combinatorics (matroids and polytopes) and geometry (projective varieties, Chow rings, and intersection cohomology), yielding new proofs and extensions of known Lefschetz phenomena and enabling polarization results for intersection cohomology via the decomposition framework. Overall, the work provides a unifying algebraic approach to the decomposition theorem’s Hodge-theoretic aspects, with implications for standard conjectures and polarized Hodge structures in geometric contexts.

Abstract

A Lefschetz module is a module over a graded algebra that satisfies analogues of Poincaré duality, the Hard Lefschetz property, and the Hodge--Riemann relations with respect to an open convex cone in the degree one part of . We analyze its decomposition into indecomposable modules over subrings of that are generated by elements in the closure of , establishing structural results that parallel the decomposition theorem for morphisms of complex projective varieties. We use our theorems to recover key statements in combinatorial Hodge theory and illuminate the Hodge-theoretic aspects of the decomposition theorem in algebraic geometry.

Paper Structure

This paper contains 20 sections, 48 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The decomposition package in practice
  3. Coefficient fields
  4. Applications to matroids
  5. Applications to polytopes
  6. Applications to projective varieties
  7. Algebraic cycles
  8. Lefschetz modules with Hodge structures
  9. The Hodge theory of the decomposition theorem
  10. Examples
  11. The perverse filtration
  12. Module structures
  13. The 1-dimensional summands
  14. The descent lemma
  15. The bilinear form on $\operatorname{Gr}$
  16. ...and 5 more sections

Key Result

Theorem 1.1

If $d(\alpha) \le d(\beta)+2k$, then

Theorems & Definitions (80)

  • Theorem 1.1: Simplicity
  • Theorem 1.2: Duality
  • Theorem 1.3: Hard Lefschetz
  • Theorem 1.4: Hodge--Riemann
  • Theorem 1.5: Relative Hard Lefschetz
  • Corollary 1.6
  • Theorem 1.7: Relative Hodge--Riemann relations
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 1.10: Decomposition
  • ...and 70 more