It is known that the toric variety $X_\Sigma$ of a simplicial fan $\Sigma$ can be constructed as a quotient $$X_\Sigma = \bigl(\mathbb C^N \setminus V(B)\bigr)/G.$$ Here $N$ is the number of rays, $B$ is the irrelevant ideal and $G$ is a certain subgroup of the big torus $(\mathbb C^\ast)^N$.
Assume that $\Sigma$ is a normal fan of a polytope. Then it is also known that
(*) $\mathbb C^N \setminus V(B)$ is the complement of the union of sets of (complex) codimension at least $2$.
Questions:
I have heard (at several places) that the fact (*) implies that $\mathbb C^N \setminus V(B)$ is simply connected. How can we prove this?
Can we see that $\pi_2(\mathbb C^N \setminus V(B))$ vanishes as well?
Notes:
- We follow the notation in Cox's lecture notes. (Lectures on toric varieties)
- If $X_\Sigma$ is smooth, we can see that $\pi_2(\mathbb C^N \setminus V(B))=0$: Since $A_{n-1}(X_\Sigma) = H_{2n-2}(X_\Sigma) \cong H^2(X_\Sigma)$ is a free abelian group of rank $b$, $G=\mathrm{Hom}_{\mathbb Z}(A_{n-1}(X_\Sigma),\mathbb C^*)$ is a complex torus of dimension $b$. The homotopy long exact sequence gives us a short exact sequence of free abelian groups $0\to\pi_2(\mathbb C^N \setminus V(B))\to\pi_2(X_\Sigma)\to\pi_2(BG)\to0$. Since $X_\Sigma$ is $\pi_2(X_\Sigma)$ is also a free abelian group of rank $b$ ($\because$ Hurewicz theorem), the homomorphism $\pi_2(X_\Sigma) \to \pi_2(BG)$ is an isomorphism.
- I am looking for a direct proof of the 2-connectedness of the Zariski open subset $\mathbb C^N \setminus V(B)$. Because if we have such a proof, then we can use it to compute some homotopy groups (and cohomology groups) conversely.
- Please do not use glueing construction of toric varieties.
If $X$ is a subvariety of $\mathbb{C}^N$ of complex codimension $\leq 2$, then the fact that $\mathbb{C}^N\smallsetminus X$ is simply connected follows from the normality of $\mathbb{C}^N$ and the Zariski-Nagata purity theorem; see SGA2 X Theorem 3.4(i).
You can get both $\pi_1$ and $\pi_2$ trivial by using general position. See Lemma 2.4 of this paper (full disclosure: I'm the 3rd author) for an argument, which you can apply to the underlying real structure.