# divisors in non-compact toric varieties

Let $$X$$ be a smooth quasi-projective toric variety of dimension $$n$$ over $$\mathbb C$$. Take it to be non-compact, so its fan is not complete. (A good example to keep in mind is a toric Calabi-Yau.)

If we denote by $$D_i$$ for $$i=1,\ldots,d$$ the irreducible subvarieties of codimension one that are stable under the torus action (corresponding to edges of the fan), what can be said about the kernel $$K$$ in the exact sequence $$0 \to K \to \oplus_{i=1}^d \mathbb Z D_i \to A_{n-1}(X) \to 0$$ in this case?

I know that if $$D_i$$ is also compact, then its Poincare dual $$PD[D_i]$$ defines a class in compact support cohomology $$H^2_c (X)$$, but I'm confused how this is related with usual $$H^2(X)$$, hence my question above.

• The kernel $K$ is just the monomial lattice $M$, and the map $M\to \bigoplus \mathbf{Z}D_i$ sends $m$ to $\operatorname{div}(m) = \sum \langle m, \rho_i\rangle \cdot D_i$ where $\rho_i\in N = M^\vee$ is the generator of the ray of the fan corresponding to $D_i$. Apr 30, 2021 at 20:40
• Strictly speaking, the kernel is the image of this map, but it is injective in most cases. A necessary and sufficient condition is that the fan is not contained in a hyperplane of $N_\mathbf{R}$. Apr 30, 2021 at 20:42
• @PiotrAchinger Thanks. Since $X$ is non-compact, the fan is contained in a hyperplane, right? if so, it is meaningful to ask for a characterization of the kernel, or more generally how to best deal with divisors in the non-compact setting.
– jj_p
May 1, 2021 at 13:33
• No, $X$ is compact if and only if the fan is complete i.e. it covers $N_\mathbf{R}$. There is plenty of fans which are not complete and not contained in a hyperplane, e.g. the fan corresponding to $\mathbf{A}^n$. May 1, 2021 at 15:17

As Piotr mentioned in the comments, you don't need completeness of the fan to be sure that the sequence $$0 \to M \to \oplus \mathbb Z D_i \to Cl(X_{\Sigma}) \to 0$$ is exact. The only condition that you actually need is that fan $$\Sigma \subset N_{\mathbb R}$$ is not contained inside (real) linear subspace of codimension 1. See Cox Little Schenck Theorem 4.1.3.
I believe you wanted to say "since $$X$$ is Calabi-Yau (necessary non-compact) the fan is contained in hyperplane" (which is true). In this case you can always write $$0 \to M/(\operatorname{ker}(M \to \bigoplus \mathbb Z D_i)) \to \bigoplus \mathbb Z D_i \to Cl(X_{\Sigma}) \to 0$$ This sequence is exact for any toric variety $$X_{\Sigma}$$ associated with fan $$\Sigma$$. And $$\operatorname{ker}(M \to \bigoplus \mathbb Z D_i)$$ is exactly the covectors which are ortohonal to the fan generators, so you can write $$0 \to M/L^{\bot} \to \bigoplus \mathbb Z D_i \to Cl(X_{\Sigma}) \to 0$$ where $$L = \operatorname{span}_{\mathbb Z} \Sigma(1)^{gen} \subset N$$ is $$\mathbb Z$$-submodule inside $$N$$ generated by ray generators of fan $$\Sigma$$ and $$L^{\bot} = \{\phi \in M | \forall l \in L : \phi(l) = 0\}$$ the annihilator.