I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties.
I wanted to know a road map for getting into combinatorial Hodge theory and other prerequisites that I'll need. I will highly appreciate if you can also recommend some good references.
My main purpose is to run a seminar that is attended by grad students and faculty members of our research group. Our end goal is to be able to learn about the work of June Huh.
Note that "combinatorial Hodge theory" can be in the context of fans (as in the work of Karu), Coxeter groups (as in the work of Elias and Williamson), or matroids. While there are certain common techniques and lemmas, there are essentially no general results in combinatorial Hodge theory. I will focus on the case of matroids.
In the setting of matroids, there are only a few papers which prove results that one could call combinatorial Hodge theory. The majority of papers in the area are studying various geometric models which are associated to matroids, and either applying known results to extend to non-realizable matroids, or using techniques that don't require combinatorial Hodge theory.
In my opinion, the most important thing is to understand the geometry of various varieties which are related to matroids. A great source for this is Eric Katz's survey Matroid theory for algebraic geometers, which is quite detailed. Unfortunately, it's from 2014, and a lot has happened since then. More up-to-date is Chris Eur's survey Essence of independence: Hodge theory of matroids since June Huh, but it is not as detailed. Chapter 2 of my thesis also contains a survey of some results about the geometry.
It is not really necessary to understand the proofs of the combinatorial Hodge theory results in order to apply them, and some of the results have no exposition other than the original papers. If one wants to learn the proofs, I would recommend reading A semi-small decomposition of the Chow ring of a matroid by Braden-Huh-Matherne-Proudfoot-Wang and ignoring the discussion of the augmented wonderful variety/Chow ring. The original paper by Adiprasito, Huh, and Katz is also very clearly written, and section 7 has a good description of the general inductive strategy which is used in all combinatorial Hodge theory papers.
