Inequality number of facets simplicial complex

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 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.

• 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" Dec 30, 2018 at 17:30
• 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 $\Delta$. Dec 30, 2018 at 17:57
• 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/… Dec 30, 2018 at 19:47
• @SamHopkins Thank you for these interesting references Dec 30, 2018 at 20:47
• (As Karim points out below, for what I said about toric varieties, the adjective projective is crucial.) Dec 30, 2018 at 22:09

$$A^\bullet(\Delta)$$ 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.
• Is $A^{\bullet}(\Delta)$ a shorthand for $A^{\bullet}(\Psi)$ where $\Psi = (\Sigma,\Delta)$ is a relative simplicial complex? Dec 30, 2018 at 22:02
• Or is $A^{\bullet}(\Delta)$ what is denoted $\mathcal{K}^{\bullet}(\Sigma,\Delta)$ in your paper? Dec 30, 2018 at 22:06
• 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. Dec 30, 2018 at 22:56
• @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? Dec 30, 2018 at 23:18