A decomposition theorem for Lefschetz modules

TL;DR

This paper introduces a decomposition theorem for Lefschetz modules, revealing their structure and connections to Hodge theory, with applications to combinatorial and algebraic geometry.

Contribution

It establishes a new decomposition theorem for Lefschetz modules, paralleling the geometric decomposition theorem and linking algebraic and combinatorial Hodge theories.

Findings

Decomposition of Lefschetz modules into indecomposables over specific subrings.

Structural results analogous to the decomposition theorem for complex projective varieties.

Applications to key statements in combinatorial Hodge theory.

Abstract

A Lefschetz module is a module over a graded algebra $A$ that satisfies analogues of Poincar\'{e} 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.