Karim in his youth with a fan

Congratulations, Karim!

**Update**: Here is the link to the paper

*From the arXive, Dec 26, 2018. (Link will be added tomorrow.)*

COMBINATORIAL LEFSCHETZ THEOREMS BEYOND POSITIVITY

by Karim Adiprasito

**Abstract:** Consider a simplicial complex that allows for an embedding into . How many faces of dimension or higher can it have? How dense can they be?

This basic question goes back to Descartes. Using it and other fundamental combinatorial

problems, we will introduce a version of the Kähler package beyond positivity,

allowing us to prove the Lefschetz theorem for toric varieties even when the ample

cone is empty. A particular focus lies on replacing the Hodge-Riemann relations by a

non-degeneracy relation at torus-invariant subspaces, allowing us to state and prove a

generalization of the theorems of Hall and Laman in the setting of toric varieties. Of

the many applications, we highlight two main applications, one because it is the most

well-known, the other because it provided the most guiding light.

(1) We fully characterize the possible face numbers of simplicial spheres, resolving the

so called *g*-conjecture of McMullen in full generality and generalizing Stanley’s

earlier proof for simplicial polytopes.

(2) We prove that for a simplicial complex *K* that embeds into , the number of *d*-dimensional simplices exceeds the number of *(d − 1)*-dimensional simplices by a factor of at most *d + 2*. This generalizes a result of Descartes, and resolves the Grünbaum-Kalai-Sarkaria conjecture.

_______

(GK:) A few further comments. Probably the *g*-conjecture for spheres is the single problem I knock my head against the most. It is great to see it settled and it is even greater to see it settled by my friend and colleague Karim Adiprasito.

To the three ingredients of the standard conjectures (See also the previous post), Poincare duality **(PD**), Hard Lefschetz (**HL**) and Hodge-Riemann (**HR**), Karim adds the **Hall-Laman relations**. Very roughly, the Hall-Laman relations substitute** (HR)** and apply genericity (rather than definiteness) toward **(HL)**.

(We still need a good acronym for Hall-Laman, maybe **(AHL)**.)

One very nice feature of Karim’s proof is that **vertex decomposable** spheres play a special role in the path toward the proof. Those were introduced by Provan and Billera in the context of the Hirsch conjecture.

We have devoted plenty of posts to the *g*-conjecture for spheres, and mentioned it in even more posts. For an introduction to the conjecture see Eran Nevo introductory post, and the post How the g-Conjecture Came About. There is also plenty left to be done beyond the g-conjecture.

Merry X-mas and Happy new year 2019 to all our readers.

Here is the link to the paper on the arXive.

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Naive question: the -conjecture (now theorem) completely classifies the possible -vectors of a -dimensional triangulated sphere. Does it also completely classify for such spheres the possible collections of for , where for is the number of chains of -dimensional face contained in an -dimensional face contained in …? I know this is somehow related to the "cd-index" but don't know the specifics.

GK: For latex write the expression (just with one dollar sign and start with the word latex. E.g.%latex f_0%replace% to $to getDear Samuel, For simplicial complexes the face numbers determine the flag numbers . In your notations, where is the flag number for for an dimensional simplex. In the non simplicial case this is a great problem.

Oh of course I see this is completely trivial in the simplicial case. Silly me.

As far as I can remember, the CD index (as other invariants) can be of interest also for simplicial complexes although it is determined by the f-vector. Similarly, toric h-vectors for simple polytopes. As a matter of fact the flag numbers are determined by face numbers even for polytopes all whose facets are simplicial but the invariants usually expressed in terms of flag numbers are still of interest.

Dear Sam, here are some posts where flag numbers CD indices etc are discussed.

The main one is: https://gilkalai.wordpress.com/2008/06/22/eulers-formula-fibonacci-the-bayer-billera-theorem-and-fines-cd-index/

Others are:

https://gilkalai.wordpress.com/2008/06/12/billerafest/ ; https://gilkalai.wordpress.com/2018/06/20/beyond-the-g-conjecture-algebraic-combinatorics-of-cellular-spaces-i/

https://gilkalai.wordpress.com/2018/06/06/a-mysterious-duality-relation-for-4-dimensional-polytopes/

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A very nice discussion over reddit: https://www.reddit.com/r/math/comments/aa1ze3/the_gconjecture_has_likely_been_proven/

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