For a cup of coffee more

This is born out of an attempt to find equivariant Lefschetz elements, so to have a combinatorial Lefschetz theorem that is a little less generic for some conjectures in geometric topology. Actually this succeeds to give some interesting results, and I will update in a second part. It will take me some days, in the meantime I made this here simple and clear.

Dear X,

so, Covid seems almost over (fingers crossed). I lost three I loved during the time (none of them due to covid, funny enough; two suicides and a heart failure), I had covid twice (fingers crossed for the hat trick) despite three vaccinations. I am in Sofia, at a inaugural conference of the ICMS.

But that’s not what this post is about. This post is about another proof of the g-conjecture. Well, the Lefschetz property for simplicial cycles, really. It is the simplest one yet, but that is not why it is important. It is also the third one (that is essentially different) and combines the ideas of three teams. (I count the original one by me here, and the characteristic two proof by Papadakis and Petrotou; our joint paper is a combination of the ideas from the former and yada yada yada creative counting to make the holy number work out. Deal with it.) If the first is a refined choreography of slashes and parries that is difficult to follow, and the second the equivalent of wooshing around with bloodhound step (using a miraculous formula that comes out of nowhere), then this is the equivalent of bonking the boss with a hammer: we write down a rational function and examine it, observing it has a pole to show it is nontrivial. Unfortunately it seems to be less general than either of the previous proofs, but I will see whether it can be pushed.

So I assume that I have a PL sphere of even dimension .

I also restrict myself to the middle Lefschetz property. That means that I want to prove, for some Artinian reduction of the face ring , any field you like, with respect to the linear system of parameters , that we have an isomorphism.



induced by multiplication with an element . Finally, this proof also only seems to work in characteristic two and zero (update: it works in general char, but is more involved. Wait for part 2)

Those are all restrictions I make here, the full argument I will write in a note soon.

Now, remember that cool property called biased pairings I told you about? So, if is a disk of dimension , and , then there is the Poincaré pairing

.

Now, as you remember, if , then the middle Lefschetz property is equivalent to saying that the pairing



is perfect (which I defined using the map ). This is the biased pairing property I discovered for my original proof, and it was the breakthrough needed to prove the Lefschetz property in the combinatorial setting. Indeed, all proofs rely on it, and it is my great pride and shame. Pride because it was then rediscovered by Kalle Karu and Stavros Papadakis and Vasso Petrotou, which affirmed it was the right way of going about the problem. Shame because “rediscovered” means that it was written in such a difficult way that they had to reinvent it.

For future reference, let us denote the identification

by . We will choose a natural normalization later.

Anyway, to continue in the text: One of the ideas to prove the biased pairing property is to use a simple decomposition of the disk . In my original paper, I used a decomposition that was rather involved, but Kalle Karu had the ingenious idea of using just a shelling of . His proof did not quite work, but I will avenge him here.

A shelling is a way of adding facets (simplices of dimension ) one by one so that I stay a disk at any point in time. In fact, I can assume that is shellable by a classical result of Pachner: Every PL sphere is the boundary of a shellable disk.

Now, we can actually give a basis of along such a shelling : a new monomial is added whenever there is a new cardinality simplex in whose boundary lies in . Here is just an index that rises whenever such a face occurs. Notice that in such a case, a facet is added to that has not appeared in any of the previous

That is the second trick. Now we turn the corresponding monomials into an orthogonal basis under the Poincaré pairing using the Gram-Schmidt process. Let us denote the orthogonalization of with respect to $ by .

Here, the final team and its idea enters: Stavros Papadakis and Vasso Petrotou suggested we look function fields, and hence not at the field , but at the field of rational functions over with indeterminates the coeffcients of . With this, for instance, it is natural to choose, for a maximal face of

Here is the sign of with respect to some chosen ordering on the vertices, and is the minor of the matrix given by the columns corresponding to .

For monomials that are not squarefree, the degree can be computed as well, as observed in the following formula of Carl Lee:

where we understand to be the matrix obtained by replacing the -th column of with any fixed vector.

The key is now to understand

Recall that . Hence

We can expand this and compute with Lee’s formula: Notice now that the only time the determinant of the minor of appears is in the expansion of , where its inverse appears. (As Kaiying Hou pointed out to me, I should remind everyone here that we restrict to characteristic 2 here, so that the mixed term in the square vanishes. Then we can assume the remaining term does not have a pole by induction on j. Thanks Kaiying)

To make this really simple, consider what happens if we have k vertices of D and move them around. Formally, pick those vertices, and consider the partial differentials of moving them into k linearly independent directions.

Claim: The composition of these partial differentials on vanishes unless the vertices form a face of D. It is nonzero on

The first part is a simple induction. The second a simple calculation.

Hence, keeping everything else in general position but degenerating that minor, the has a pole, so the rational function is not trivial. But this means that the orthogonalization leads to a matrix for the biased pairing that is diagonal with nontrivial diagonal entries . Hence the matrix is nondegenerate, and the pairing perfect. Done.

Concluding Remarks

Well why did we make this effort? For that you will have to be a little patient. In short, it turns out that this new method allows for combinatorial Lefschetz elements that are more restricted than just generic, and in particular those that are equivariant under certain group actions. Why is that important? Well, the key lies in the book of Gromov, psalm 137.

An intriguing question that remains open is to show full anisotropy for general simplicial cycles. I know exploited that I could show biased pairing, that is, nondegeneracy of the Poincaré pairing at an ideal (namely ). But Papadakis and Petrotou (and then with me) showed in characteristic two the Poincaré pairing degenerates at no ideal in a Gorenstein ring; that seems more tricky. I will post some more detailed conjectures soon, and go into quadratic forms on function fields. Cool stuff.

Finally let me announce a theorem that is actually new, and is joint work with Stavros Papadakis, (my future postdoc) Vasso Petrotou: we are now able to push the Lefschetz principle beyond combinatorial settings (using yet another method), and prove in particular

Theorem The vector of an IDP Gorenstein lattice polytope is unimodal.

See you all soon, Love, Karim