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One of the most useful tools in the study of convex polytope is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on the combinatorics of the polytopes. This construction requires that the polytope is rational which is a real restriction when the polytope is general (neither simple nor simplicial). Often we would like to consider general polytopes and even polyhedral spheres (and more general objects) where the toric variety construction does not work.

I am aware of very general constructions by M. Davis, and T. Januszkiewicz, (one relevant paper might be Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) and several subsequent papers). Perhaps these constructions allow you to start with arbitrary polyhedral spheres and perhaps even in greater generality.

I ask about an explanation of the scope of these constructions, and, in simple terms as possible, how does the construction go?

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Gil, there is our construction with Kollar, for Euclidean and hyperbolic polyhedral complexes (it will work for spherical complexes as well, we just did not find any use for it), see arxiv.org/abs/1109.4047 and arxiv.org/abs/1201.3129. Unfortunately, at least for now, its main application is from polyhedral side to algebro-geometric side, not in the reverse direction that you are interested in. The construction almost inevitably produces singular projective varieties, however, and this is its main point, one can control the singularities. –  Misha Jun 14 '12 at 12:29
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4 Answers

The construction of Davis and Januskiewicz can be realized as an equivariant colimit.

Let $P$ be a simple polytope of dimension $n$ and let $G$ be either the mod 2 torus ${\mathbb{Z}}_2^n$ or the usual torus ${\mathbb{T}}^n$. A characteristic function on $P$ corresponds to a order-reserving map $\chi:{\mathrm{Face}} \, P\to {\mathrm{Sub}}_{\mathbf{Grp}} G$ from the face poset of $P$ to the poset of subgroups of $G$ such that

  1. The image of $\chi$ lands in the unimodular subgroups of G.
  2. $\chi$ is graded, in the sense that ${\mathrm{codim}} \,F = {\mathrm{rank}} \, \chi F$.

There is a functor $-\times G/-:{\mathrm{Face}} P\times {\mathrm{Sub}}_{\mathbf{Grp}} G\to {\mathbf{Top}}G$ from the poset product to the category of $G$-spaces that carries $(F,H)$ to the $G$-space $F\times G/H$. Here $G$ acts on the second factor of the product naturally: $$(x,Hg)g':=(x,Hgg')$$

Pick out a certain subposet $Q$ of ${\mathrm{Face}} P\times {\mathrm{Sub}}_{\mathbf{Grp}} G$ by requiring that $(F,G/H)$ is in $Q$ if and only if $H$ is a unimodular subgroup of $\chi F$. The real or complex quasitoric manifold $M(\chi)$ over $P$ with characteristic function $\chi$ is the colimit of the composite $$Q\hookrightarrow {\mathrm{Face}} P\times {\mathrm{Sub}}_{\mathbf{Grp}} G\xrightarrow{-\times G/-} {\mathbf{Top}}_G$$
Without reference to morphisms and wrting $H\prec \chi F$ to mean that $H$ is a unimodular subgroup of $\chi F$, we can write $$M(\chi)={\mathrm{colim}}_{H \prec \chi F} F\times G/H$$

Replacing the face poset by a general poset and taking a general topological group $G$ (or Lie group), one can construct $G$-spaces via such an equivariant decomposition. The defect of such a generalization is that it is unclear whether the resulting $G$-space has a manifold, variety or Kahler structure.

Gil, does a polyhedral sphere have a naturally associated poset over which we could construct $G$-spaces with interesting combinatorial invariants?

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Many thanks Colin! (I don't know the answer to your question.) –  Gil Kalai Nov 12 '13 at 18:32
    
Is there a natural poset associated to a polyhedral spheres? –  Colin Tan Nov 13 '13 at 15:07
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Dear Gil, in addition to Dan's answer let me mention that the construction of toric varieties à la Cox has been generalized to arbitrary convex polytopes in Geometric spaces from arbitrary convex polytope, Int. J. Math., 23, (2012) (the simple case had been treated in a previous paper joint with E. Prato).

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Many thanks, Fiammetta! I wonder if we can do something for polyhedral spheres which are neither polytopes or simplicial or simple. Even polhedral 3-spheres... –  Gil Kalai Mar 24 '13 at 16:06
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Dear Gil, we have a nonrational construction with F. Battaglia in the case of simplicial fans here: arXiv:1108.1637, where we use foliated compact manifolds instead of toric varieties. In this setting, Stanley's proof of (the necessary part of) the g-conjecture carries over.

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Many thanks, Dan! –  Gil Kalai Mar 12 '13 at 15:14
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The DJ construction works with a simplicial complex $K$ and a subtorus $W\leq\prod_{v\in V}S^1$ (where $V$ is the set of vertices of $K$). People tend to be interested in the case where $|K|$ is homeomorphic to a sphere, but that isn't really central to the theory. However, it is important that we have a simplicial complex rather than something with more general polyhedral structure. It is also important that we have a subtorus, which gives a sublattice $\pi_1(W)\leq\prod_{v\in V}\mathbb{Z}$, which is integral/rational information. I don't think that the DJ approach will help you get away from the rational case.

I like to formulate the construction this way. Suppose we have a set $X$ and a subset $Y$. Given a point $x\in\prod_{v\in V}X$, we put $\text{supp}(x)=\{v:x_v\not\in Y\}$ and $K.(X,Y)=\{x:\text{supp}(x) \text{ is a simplex}\}$. The space $K.(D^2,S^1)$ is a kind of moment-angle complex, and $K.(D^2,S^1)/W$ is the space considered by Davis and Januskiewicz; it has an action of the torus $T=\left(\prod_{v\in V}S^1\right)/W$. Generally we assume that $W$ acts freely on $K.(D^2,S^1)$. There is a fairly obvious complexification map $K.(D^2,S^1)/W\to K.(\mathbb{C},\mathbb{C}^\times)/W_{\mathbb{C}}$. Under certain conditions relating the position of $W$ to the simplices of $K$, one can check that $K$ gives rise to a fan, that the complexification map is a homeomorphism, and that both $K.(D^2,S^1)/W$ and $K.(\mathbb{C},\mathbb{C}^\times)/W_{\mathbb{C}}$ can be identified with the toric variety associated to that fan.

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Thanks, Neil! Are there extensions of this construction to more general type of complexes? –  Gil Kalai Jun 14 '12 at 11:01
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