This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference.
Here are two examples of the kind of combinatorial abstractions of geometric objects referred to in the title of this question:
- Coxeter groups. These are abstractions of Weyl groups. Weyl groups have geometry coming from Lie theory: they are finite reflection groups associated to a crystallographic root system. Weyl groups (or perhaps finite reflection groups, or including Weyl groups associated to affine lie algebras, etc.) are then the "realizable" Coxeter groups.
- Matroids. These are abstractions of collections of vectors in some vector space. The matroids coming from collections of vectors in some vector space (over some field, say) are again the "realizable" matroids.
Here is what I mean by "behave so well":
Often it happens that we can associate some interesting polynomial invariant to the combinatorial object in question. Some examples are:
- The Kazhdan-Lusztig (KL) polynomial associated to a Coxeter system.
- The characteristic polynomial associated to a matroid.
- The recent KL polynomial associated to a matroid (see Elias, Proudfoot, and Wakefield – The Kazhdan-Lusztig polynomial of a matroid).
And these polynomials have surprising and deep properties (positivity or unimodality/log-concavity of coefficients) that are not at all obvious from their definitions. A recurring theme is that these properties can be established in the "realizable" cases by appealing to algebraic geometry, specifically, to some suitable cohomology theory. However, the properties continue to hold for the general, nonrealizable objects for which there is no underlying geometry. The proofs of the general result are usually more "elementary" in so far as they avoid any algebraic geometry; but chronologically they come after the realizable results.
For instance, the coefficients of KL polynomials associated to a Coxeter system are positive. This was a famous conjecture of Kazhdan–Lusztig, proved a few years ago by Elias and Williamson (The Hodge theory of Soergel bimodules). However, positivity was known for realizable Coxeter groups much earlier by interpreting the polynomials as Poincaré polynomials for the intersection cohomology of certain Schubert varieties.
Similarly, it is conjectured that the KL polynomial of a matroid has positive coefficients (see Gedeon, Proudfoot, and Young – Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures); and this conjecture is known to be true when the matroid is realizable, again by interpreting the coefficients as dimensions of intersection cohomology spaces on certain varieties.
Or for the characteristic polynomial of a matroid: we know that the coefficients of this polynomial are log-concave, as was recently proved in the remarkable work of Adiprasito–Huh–Katz (Hodge Theory for Combinatorial Geometries). Again, this result was preceded by the same result for the realizable case, due to Huh–Katz (Hodge Theory for Combinatorial Geometries), interpreting the coefficients as intersection numbers for some toric variety.
So we come to my question:
Why do combinatorial abstractions of geometric objects behave so well, even in the absence of any underlying geometry?
EDIT: At around the 50 minute mark of his plenary talk at ICM 2018 (on Youtube here: Representation theory and geometry), Geordie Williamson asks a roughly similar question, and suggests that it may be a "mystery for the 21st century."
EDIT 2: As mentioned in the answers of Gil Kalai and Karim Adiprasito, another good example of "combinatorial abstraction of geometric object" is the notion of simplicial sphere, where the realizable case is a boundary of a polytope. Here the realizable case is connected to algebraic geometry via the theory of toric varieties, and as always this connection enables one to prove deep positivity results (e.g. the g-theorem of Stanley); whereas again the same results for the nonrealizable case are apparently much harder and the subject of intense, current research.
EDIT 3: I'm including a very relevant passage from a preprint of Braden-Huh-Matherne-Proudfoot-Wang (Singular Hodge theory for combinatorial geometries).
Remark 1.13 It is reasonable to ask to what extent these three nonnegativity results can be unified. [The three results here are the nonnegativity of the coefficients of the KL polynomial of an arbitrary Coxeter group, the $g$-polynomial of an arbitrary polytope, and the KL polynomial of an arbitrary matroid.] In the geometric setting (Weyl groups, rational polytopes, realizable matroids), it is possible to write down a general theorem that has each of these results as a special case. However, the problem of finding algebraic or combinatorial replacements for the intersection cohomology groups of stratified algebraic varieties is not one for which we have a general solution. Each of the three theories described above involves numerous details that are unique to that specific case. The one insight that we can take away is that, while the hard Lefschetz theorem is typically the main statement needed for applications, it is always necessary to prove Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations together as a single package.
EDIT 4: Another really good summary of this sea of ideas is given in Huh's ICM 2022 article "Combinatorics and Hodge theory", especially the introduction section. Quoting from there:
The known proofs of the Poincaré duality, the hard Lefschetz property, and the Hodge–Riemann relations for the objects listed above have certain structural similarities, but there is no known way of deducing one from the others. Could there be a Hodge-theoretic framework general enough to explain this miraculous coincidence?
Thus there is not yet a satisfactorily general answer to this question (but maybe one day there will be one).
Best Answer
Perhaps this, for now, is more an issue of perspective. Yes, for matroids, spheres and Coxeter groups the realizable cases were known before using results in algebraic geometry, but this is natural as our understanding of the cohomology of algebraic varietes was much better, historically. And so we think of this as strange because we are used to think of this in terms of algebraic varieties.
However, matroids, for instance, are perhaps more naturally thought of in the context of valuations, and there, it suddenly becomes more natural to consider McMullen's argument for the Lefschetz theorem and the Hodge-Riemann relations (and this is ultimately what is used).
Similarly, spheres are rarely ever polytopal, and even for those that are, the realization as a polytope is an unnatural straightjacket. We do, however, understand them well in terms of cobordisms, and we do know general position tricks from when we define intersection products in cohomology. And this ultimately leads to the Lefschetz theorem there.