Recall that for a cone \(\sigma\) in a polyhedral fan \(\Sigma\) in a lattice \(N\) with dual lattice \(M\),
- \(N_\sigma\) denotes the linear span of \(\sigma\cap N\)
- \(M(\sigma) = \sigma^\perp \cap M\) is the character lattice of the toric variety given by the orbit closure \(V(\sigma)\) in \(X_\Sigma\).
Theorem
Let \(\Sigma\) be a polyhedral fan in a lattice \(N\cong \mathbb Z^n\) and \(d = \dim X_\Sigma\). Denote by \(V(\sigma)\) the torus orbit closure corresponding to the cone \(\sigma \in \Sigma\). Then the \(k\)th Chow group of \(X_\Sigma\) is generated by the orbit closures \(\{V(\sigma)\}_{\sigma \in \Sigma(n - k)}\) and has the following presentation: \[\bigoplus_{\tau\in \Sigma(n - k - 1)}M(\tau) \xrightarrow{\alpha} \bigoplus_{\sigma \in \Sigma(n - k)} \mathbb Z \cdot [V(\sigma)] \to A_k(X_\Sigma) \to 0.\] The map \(\alpha = \bigoplus \alpha_\tau\) is defined on \(m \in M(\tau)\) by \[\alpha_\tau(m) = \sum_{\sigma \in \Sigma(n - k)}\langle m, u_{\sigma, \tau}\rangle \cdot [V(\sigma)]\] where \(u_{\sigma,\tau}\) is the unique element in the lattice \(N_\sigma/N_\tau \cong \mathbb Z\) which is
- a generator for \(N_\sigma/N_\tau\) as a group (of which there are two choices) and
- is represented by an integral point of \(\sigma\).