Definition
Let \(B\) be an integral affine manifold with singularities. A polyhedral decomposition of \(B\) is roughly a compatible collection of polyhedral decompositions of the charts of \(B\). More precisely, it is a collection \(\mathcal P\) of closed subset of \(B\) called cells covering \(B\) which satisfies the following properties. First, we require that all vertices of \(\mathcal P\) are nonsingular, that is, if \(\{v\} \in\mathcal P\) then \(v\in B\setminus \Delta\). Given such a \(v\), we then require that there exists a closed neighborhood \(R_v\subseteq T_{B,v} \) of the origin at the tanget space of \(B\) at \(v\) with a polyhedral decomposition \(\mathcal P_v\) together with the continuous map \(\exp_v:R_v\to \exp_v(0) = v\) (that is, \(R_v\) is a “closed chart” of \(B\) at \(v\), with the origin taking the place of \(v\), and \(\mathcal P_v\) is a polyhedral decomposition of \(R_v\) in the usual sense for a closed subset of Euclidean space). We additionally require the following:
- \(\exp_v\) is locally a homeomorphism onto its image, is injective on \(\Int(\tau)\) for all \(\tau\in \mathcal P_v\), and is an integral affine map in some neighborhood of the origin.
- For every top-dimensional \(\widetilde \sigma \in \mathcal P_v\), \(\exp_v(\Int(\widetilde \sigma)) \cap \Delta = \emptyset\) and the restriction of \(\exp_v\) to \(\Int(\widetilde \sigma)\) is integral affine. Furthermore, \(\exp_v(\widetilde \tau) \in \mathcal P\) for all \(\widetilde \tau \in \mathcal P_v\).
- A cell \(\sigma\in \mathcal P\) contains \(v\) if and only if \(\sigma = \exp_v(\widetilde \sigma)\) for some \(\widetilde\sigma \in \mathcal P_v\) for all \(\widetilde \tau \in \mathcal P_v\).
- Every \(\sigma\in \mathcal P\) arises in this way, that is, every \(\sigma\) contains a vertex \(v\) with \(\{v\}\in \mathcal P\).
We have an additional condition which will often be of interest, and we say that \(\mathcal P\) is toric if it satisfies it:
- For each \(\sigma\in \mathcal P\), there is a neighborhood \(U_\sigma\subseteq B\) of \(\Int(\sigma)\) and an integral affine submersion \(S_\sigma:U_\sigma \to M' _ {\mathbb R}\) where \(M'\) is a lattice of rank equal to \(\dim B - \dim \sigma\) and \(S _\sigma(\sigma\cap U_\sigma) = \{0\}\).