algebraic geometry

Just a quick post that kills some questions from the previous post, and a report on just an overall wonderful REU project summer.

So let me start by the following: myself, and then Papadakis and Petrotou, and then us three jointly, proved the Lefschetz property for triangulated spheres (characteristic two in the case of Papadakis and Petrotou). This may be an unfamiliar word for someone not from algebraic geometry, but essentially, it is a property stating that Poincaré duality in certain manifolds coming from algebraic geometry (for instance, smooth projective varieties) is realized in a concrete way. Now you may think combinatorial Hodge theory à la Rota conjecture I proved with June and Eric, or positivity of Kazhdan-Lustig polynomials established by Elias and Williamson, this is much better. Because while those arguments relied on a known combinatorial trick by McMullen/de Cataldo-Migliorini, this one used an entirely new idea. First, let me state the theorem, without getting too technical:

We proved, given a triangulated sphere $\Sigma$ of dimension $d-1$ , the face ring (in arbitrary characteristic) permits an Artinian reduction and contains a linear element $\ell$ so that

$A^k(\Sigma) \xrightarrow{\ell^{d-2k}} A^{d-k}(\Sigma)$

is an isomorphism.

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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.

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