Combinatorial Hodge Theory

Developed classically in the context of complex geometry, Hodge theory gives certain rigid structures on the cohomology rings of compact Khler manifolds.In the past few decades, it has been discovered that these structures exist in other contexts as well.Finding such "Hodge theoretic" structures associated to combinatorial objects has led to remarkable developments, including resolutions of longstanding open problems in combinatorics.Here, we give a broad (and necessarily incomplete) snapshot of these developments in combinatorial Hodge theory.Previous surveys on this topic, with more extensive lists of references, can be found at [Huh23,Eur24].In Section 1, we define the structure colloquially known as the "Khler package" and discuss some general principles for its combinatorial applications.These general principles are illustrated in concrete examples in Sections 2, 3, and 4.These sections each feature a different combinatorial object, but they follow a common template outlined in Section 1.3, so the readers may pick and choose to their taste.In Section 5, we outline some strategies common to the proofs of the "Khler package" for many different combinatorial structures. The Khler PackageLet us begin with a toy example.Example 1.Let be the polynomial ring [, , ] modulo the ideal 2 , 2 , 2 .For each , the degree graded component , as a -vector space, has a basis (as an vector space) given by the square-free monomials:While the symmetry in the dimensions of the graded components is apparent, there are more refined structures.For