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In a recent preprint, Adiprasito proves that if $\Delta$ is a simplicial complex of dimension $d$ that can be embdedded in a $2d$-dimensional homology sphere (say $\Sigma$) that satisfies a version of the hard Lefschetz theorem, then: $$ f_{d}(\Delta) \leq (d+2)f_{d-1}(\Delta),$$ where $f_{k}(\Delta)$ is the number of faces of dimension $k$ of $\Delta$. This can be seen as higher dimensional analogue of Euler's inequality for a simple connected planar graph $\mathcal{C}$:

$$f_{1}(\mathcal{C}) \leq 3f_{0}(\mathcal{C})-6.$$

As far as I understand, the equality proved by Adiprasito is a version of the hard Lefschetz theorem for an analogue of the simplicial homology of the pair $(\Sigma, \Delta)$. Indeed, denote by $A_{\bullet}(X, \mathbb{Z})$ this analogue of the simplicial homology of a simplicial space.

Adiprasito explains in his preprint that we have the inequalities (or at least this what I have understood, I might be mistaken)

_$\dim A_{d}(\Delta, \mathbb{Z}) \leq f_{d-1}(\Delta)$, because the $d$-chains are $(d-1)$-dimensional faces.

_$\dim A_{d+1}(\Delta, \mathbb{Z}) \geq f_{d}(\Delta) - (d+1)f_{d-1}(\Delta)$, because $(d+1)$-chains are $d$-dimensional faces and (by construction of the simplicial complex?) each $(d-1)$-dimensional face gives exactly $d+1$ relations.

Now, assume there is an isomorphism $l : A_{d+1}(\Sigma, \mathbb{Z}) \longrightarrow A_{d}(\Sigma, \mathbb{Z})$, which respects $A_{\bullet}(\Delta, \mathbb{Z})$. If the map $A_{d+1}(\Delta, \mathbb{Z}) \longrightarrow A_{d+1}(\Sigma, \mathbb{Z})$ is injective, then, a simple diagram chasing, shows that $l$ induces an injection: $$ l : A_{d+1}(\Delta, \mathbb{Z}) \longrightarrow A_{d}(\Delta, \mathbb{Z}).$$ And we get the desired inequality.

I have a a few questions about this very nice argument:

_I have some troubles to understand concretly what is $A(\Delta)_{\bullet}$. Any suggestions?

_Adiprasito goes on proving that there are some (many?) PL-spheres that do satisfy the hard Lefschetz theorem by construction some explicit Lefschetz maps. I must say that this construction is quite involved for me and I am not able to follow it. Are there some simple examples of spheres (even for the standard one, I will be happy to understand) for which the Lefschetz map can be constructed explicitely (and relatively easily)?

_Is there a general argument to prove that the map $A_{d+1}(\Delta, \mathbb{Z}) \longrightarrow A_{d+1}(\Sigma, \mathbb{Z})$ is injective? I might be totally mistaken on this last point at is it seems to be quite the opposite of the weak Lefschetz theorem in algebraic geometry.

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    $\begingroup$ I think $A(\Delta)_{*}$ is just the Stanley-Reisner ring modulo a linear system of parameters coming from an embedding of the simplicial complex into Euclidean space. This is a standard thing to consider in this area, at least since Stanley's introduction of commutative algebraic techniques. The classic reference is Stanley's "Combinatorics and Commutative Algebra" $\endgroup$ – Sam Hopkins Dec 30 '18 at 17:30
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    $\begingroup$ As for an "explicit" construction of the Lefschetz map: in the case of e.g., the cohomology of a toric variety, it is induced from the cohomology class of a hyperplane section. If you look at Theorem 1.4 of the aforementioned monograph of Stanley this element is $x_1 + x_2 +\cdots + x_n$, corresponding to the sum of all the vertices of $\Delta$. $\endgroup$ – Sam Hopkins Dec 30 '18 at 17:57
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    $\begingroup$ Finally, Gil Kalai's blog has many posts dedicated to the g-conjecture and related topics. You can start by checking out this recent post where the result of Adiprasito is briefly discussed, and following the links there: gilkalai.wordpress.com/2018/12/25/… $\endgroup$ – Sam Hopkins Dec 30 '18 at 19:47
  • $\begingroup$ @SamHopkins Thank you for these interesting references $\endgroup$ – Libli Dec 30 '18 at 20:47
  • $\begingroup$ (As Karim points out below, for what I said about toric varieties, the adjective projective is crucial.) $\endgroup$ – Sam Hopkins Dec 30 '18 at 22:09
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I follow your notation rather than mine.

$A^\bullet(\Delta)$ is obtained as follows: You construct a linear system of parameters for $\Sigma$. If $\Sigma$ is of dimension $d-1$, then this is of length $d$. In fact, you can think of this linear system as a set of coordiantes for the vertices of $\Sigma$ in $\mathbb{R}^d$. You restrict that linear system to the face ring for $\Delta$. Note that this is in general MORE that the linear system of parameters for $\Delta$: If $\Delta$ is of dimension $k-1$, you have $d-k$ linear forms to many. In general, I think of simplicial complexes as coming with some coordinates, making this well-defined.

In particular, the surjection $$A^\bullet(\Sigma) \rightarrow A^\bullet(\Delta)$$ (or dually the injection you write down) just follows from definition.

As for constructing Lefschetz maps, Sam is right for the classical case of projective toric varieties. My construction of Lefschetz elements for general simplicial spheres is more involved, but the case of vertex decomposable spheres is perhaps understandable in an easier way.

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  • $\begingroup$ Is $A^{\bullet}(\Delta)$ a shorthand for $A^{\bullet}(\Psi)$ where $\Psi = (\Sigma,\Delta)$ is a relative simplicial complex? $\endgroup$ – Sam Hopkins Dec 30 '18 at 22:02
  • $\begingroup$ Or is $A^{\bullet}(\Delta)$ what is denoted $\mathcal{K}^{\bullet}(\Sigma,\Delta)$ in your paper? $\endgroup$ – Sam Hopkins Dec 30 '18 at 22:06
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    $\begingroup$ no, $A(\Delta)$ is face ring of $\Delta$ modulo ideal generated in degree one induced by its coordinates, in this context, its coordinates as a subcomplex of $\Sigma$. Think of it as coming with this ideal, or better yet, as any complex coming with some choice of coordinates. $\endgroup$ – Karim Adiprasito Dec 30 '18 at 22:56
  • $\begingroup$ @KarimAdiprasito Thank you for your answer. Let stick to the case of the fan of a projective variety. Assume that I don't know the hard Lefschetz theorem in algebraic geometry. Can I prove the hard Lefschetz for $x_1+ \ldots + x_n$ using your perturbation lemma 6.1? $\endgroup$ – Libli Dec 30 '18 at 23:18
  • $\begingroup$ strangely enough, no. Even nice smooth varieties like the product of two projective lines are not good enough to work with my way of proving Lefschetz. But another variety with the same equivariant cohomology (i.e. face ring) is. See Example 5.9 in my paper. $\endgroup$ – Karim Adiprasito Dec 30 '18 at 23:35

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