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From Notices of the AMS

Combinatorial Hodge Theory

Visualization of the Kähler package

by Christopher Eur
Matt Larson
Communicated by Han-Bom Moon

Developed classically in the context of complex geometry, Hodge theory gives certain rigid structures on the cohomology rings of compact Kähler 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 long-standing 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 "Kähler 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 "Kähler package" for many different combinatorial structures.

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