Stavros Argyrios Papadakis, Vasiliki Petrotou, and Karim Adiprasito

In 2018, I reported here about Karim Adiprasito’s proof of the g-conjecture for simplicial spheres. This conjecture by McMullen from 1970 was considered a holy grail of algebraic combinatorics and it resisted attacks by many researchers over the years.

Today’s post reports on two developments. The first from December 2020 is a second proof for the -conjecture for spheres by Stavros Argyrios Papadakis and Vasiliki Petrotou. (Here is a link to the arXive paper.) A second proof to a major difficult theorem is always very very important and exciting. The proof seems considerably simpler than Karim Adiprasito’s proof. The miracle that enables simplification was working with *characteristic two* in contrast to Adirasito’s proof which is characteristic free, and which therefore applies in greater generality.

However, the two teams joined forces, and in January 2021 posted a proof (which seems even simpler) by Karim Adiprasito, Stavros Argyrios Papadakis and Vasiliki Petrotou of the -conjecture largely extended to pseudomanifolds. (Here is a link to the arXived paper.)

I was too slow to report on these developments separately, so let me report here on both. I am thankful to Karim Adiprasito for telling me about both results and some further conversations.

I hope that with these new proofs more people in the community will be able to absorb and understand the ideas, go carefully over the proofs, and apply the methods to a large number of outstanding problems beyond the -conjecture. (A few of which are discussed in this post.)

## December 2020: Stavros Argyrios Papadakis and Vasiliki Petrotou present a second proof of the g-conjecture.

A few weeks ago Stavros Argyrios Papadakis and Vasiliki Petrotou uploaded a paper on the arXive The characteristic 2 anisotropicity of simplicial spheres. The paper presents a second proof for McMullen’s -conjecture for simplicial spheres.

Stavros Argyrios Papadakis and Vasiliki Petrotou: The characteristic 2 anisotropicity of simplicial spheres

**Abstract:** Assume is a simplicial sphere, and is a field. We say that is generically anisotropic over if, for a certain purely transcendental field extension of , a certain Artinian reduction of the Stanley-Reisner ring has the following property: All nonzero homogeneous elements of of degree less or equal to have nonzero square. We prove, using suitable differential operators, that, if the field has characteristic 2, then every simplicial sphere is generically anisotropic over . As an application, we give a second proof of a recent result of Adiprasito, known as McMullen’s -conjecture for simplicial spheres. We also prove that the simplicial spheres of dimension 1 are generically anisotropic over any field .

(GK) The -conjecture is about relations between face numbers of simplicial spheres. The new proof, like Karim’s proof and most earlier attacks goes through an algebraic property, the **Lefschetz property**. And here this property is proved for fields of coefficients of characteristic two. The proof relies on a formula whose discovery was assisted by computer experimentation using the Grayson-Stillman computer algebra program Macaulay II.

## January 2021: Karim Adiprasito, Stavros Argyrios Papadakis, and Vasiliki Petrotou present a new proof for the *g*-theorem extended to general pseudomanifolds, cycles, and doubly-Cohen Macualay complexes.

A few days ago Karim, Stavros Argyrios, and Vasiliki uploaded a new joint paper with yet a further simplification of the argument and far-reaching extensions to pseudomanifolds, cycles, and doubly Cohen Macaulay complexes.

Karim Adiprasito, Stavros Argyrios Papadakis and Vasiliki Petrotou, Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles

**Abstract:** We prove the hard Lefschetz property for pseudomanifolds and cycles in any characteristic with respect to an appropriate Artinian reduction. The proof is a combination of Adiprasito’s biased pairing theory and a generalization of a formula of Papadakis-Petrotou to arbitrary characteristic. In particular, we prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a generalization of the -conjecture due to Stanley. We also provide a simplified presentation of the characteristic 2 case of the Papadakis-Petrotou formula, and generalize it to pseudomanifolds and cycles.

## A few more comments on both papers

A crucial ingredient in these works is the need to understand the behaviour of Stanley-Reisner ring w.r.t. generic systems of parameters. Finding the right tools to use “genericity” is a crucial question. Adiprasito considered in his 2018 paper the **Hall-Laman property** which is related to the connection with framework rigidity and the notion of generic rigidity. This property is crucial (under somewhat smaller generality) in the newest APP paper. It is also closely related to the generic anisotropicity property studied by Papadakis and Vasiliki Petrotou.

Exploring the use of genericity and the connections with rigidity in the study of face numbers go back to works from the 80s and 90s by Stanley, Kalai, Björner and Kalai, Billera, Whitely, Lee, Tay and Whitely, and others. The reference to Hall is based on Hall’s famous marriage theorems (that has some roots also in Jacobi’s work) and its connections with generic determinants that were considered in the works of Tutte, Edmonds, Lovàsz, Rabin and Vazirani, Karp, Upfal, and Wigderson, and others. The reference to Laman refers to Laman’s 1970 theorem that gives a characterization for planar generic rigidity (we discussed it briefly in this post on derandomization).

Another crucial idea both in Adiprasito’s 2018 proof and the Papadakis- Petrotou proof is the nondegeneracy (in a strong sense) of a Poincaré pairing. Both proofs use the property that the Poincaré pairing does not degenerate at ideals (something Karim calls the **biased pairing property**) as well infinitesimal deformations of the linear system of parameters. Karim told me that both proofs, however, face distinct difficulties: while his proof is geometrically difficult, and the Papadakis- Petrotou paper is restricted by the characteristic of the field. Through combining, they managed to circumvent both these difficulties!

While, the Lefschetz property is extended to arbitrary characteristics and to very general classes of simplicial complexes, the generic anisotropicity property itself is still open beyond characteristic two. The new paper also offers a new understanding of the formula that was achieved originally by Stavros and Vasiliki with computer experimentation, and relates it to old results by Carl Lee.

I am quite happy that the new proof extends to the exciting uncharted territory of pseudomanifolds, as there are a lot we want to understand regarding their Stanley-Reisner rings. For example, it is a long dream of mine to be able to identify Goresky-MacPherson’s intersection cohomology of a simplicial pseudomanifold , (for arbitrary perversities) through the Stanley-Reisner ring.

For earlier posts on the g-conjecture click here. See especially the guest posts by Eran Nevo.

## Hilda Geiringer

An interesting historical comment is that Laman’s 1970 theorem about generic planar rigidity was proved already in 1927 by the famous applied mathematician and statistician Hilda Pollaczek-Geiringer. Here is an English exposition of her proof and some of her other works on rigidity by Brigitte Servatius.

Why are you not mentioning this paper

https://arxiv.org/abs/2001.06594

Of course you are right, one should always look at all claims and read them, and too few are willing to do it.

However, I am personally skeptical about that paper. The author uses a well-known and often attempted idea, but it offers few new ideas. But I sat down and read the paper, and here is a specific counterexample:

Essentially, the author claims that under a certain condition, bistellar moves preserve the Lefschetz property. That condition is essentially anisotropy of face monomials, that is, for a monomial m supported on a face, m^2 is not 0 (actually, the author has a different condition, which implies this, and from here he deduces the theorem).

Unfortunately, that condition is not sufficient, and there are counterexamples. See for instance the example in Section 4.5 of arxiv:1812.10454 on Bad Artinian reductions.

Feifei Fan’s strategy for attempting to prove the -conjecture by induction based on flips is certainly worth discussion and it will be great if such a strategy can prevail. Karim raised some specific concerns regarding the details.

Indeed, that would be marvellous if flips could be made to work. I am somewhat skeptical it can be done, as the preservation of the Lefschetz property is (stably) equivalent to the preservation of the biased pairing property, which can happen to be very nonlocal in general.

One of the open questions is for instance whether one can prove the Lefschetz property work with a linear system coming from the moment curve.

Another, perhaps optimistic conjecture is that for a neighborly sphere, any Artinian reduction has the Lefschetz property. There is a specific reason this might be true, as for instance the Kronecker/perturbation implies that it is partially true (i.e. for some, but not for all the Lefschetz isomorphisms).