Lemma 11.16.2.8C1AF91F05D6 Let $U$ be a connected smooth affine variety and let $q\geq 0$ be an integer. Put $B=U\times \mathbb C^*\times \mathbb C^ q$, with coordinates $(x,t,s)$, and for $\lambda \in \mathbb C^*$ let $d_\lambda (x,t,s)=(x,\lambda t,\lambda s)$. For each $j\geq 1$, let $C_ j\subset B$ be a holomorphically convex compact set, let $g_ j$ be holomorphic near $C_ j$, and let $\delta _ j>0$. Suppose that $t(C_ j)$ lies in the disk $|t-t_ j|\leq |t_ j|/4$ for some $t_ j\neq 0$. Then there are $\lambda _ j\in \mathbb C^*$ and $f\in \mathcal O(B)$ such that
\[ \sup _{d_{\lambda _ j}(C_ j)} |f-\lambda _ j g_ j\circ d_{\lambda _ j}^{-1}|<|\lambda _ j|\delta _ j \qquad (j\geq 1). \]
Proof.
Choose an exhaustion $(L_ j)_{j\geq 0}$ of the affine, hence Stein, manifold $B$ by holomorphically convex compact sets with $L_ j\subset \operatorname {int}L_{j+1}$. We construct $f_ j\in \mathcal O(B)$ and compact sets $K_ j$ inductively, starting with $f_0=0$ and $K_0=L_0$.
At stage $j\geq 1$, choose $M>0$ with $t(K_{j-1})\subset \{ |t|\leq M\} $, and choose $R>4M/3$. Dilation by $\lambda _ j=R/t_ j$ moves the $t$-projection of $C_ j$ into the disk $|t-R|\leq R/4$, which is disjoint from $|t|\leq M$. For the rest of the induction we replace $C_ j$, $g_ j$ and $\delta _ j$ by the dilated compact set $d_{\lambda _ j}(C_ j)$, the function $\lambda _ jg_ j\circ d_{\lambda _ j}^{-1}$ and the tolerance $|\lambda _ j|\delta _ j$, keeping the same notation. Lemma 11.16.1, applied with the function $t$, makes $K_{j-1}\cup C_ j$ holomorphically convex.
On disjoint neighborhoods of $K_{j-1}$ and $C_ j$, prescribe $f_{j-1}$ and $g_ j$, respectively. Oka–Weil approximation gives $f_ j\in \mathcal O(B)$ with
\[ \sup _{K_{j-1}}|f_ j-f_{j-1}|<\epsilon _ j, \qquad \sup _{C_ j}|f_ j-g_ j|<\epsilon _ j, \qquad \epsilon _ j=2^{-j-2}\min (1,\delta _1,\ldots ,\delta _ j). \]
Let $K_ j$ be a member of the fixed exhaustion containing $K_{j-1}\cup C_ j\cup L_ j$ in its interior. The interiors of the sets $K_ j$ exhaust $B$, and the summable errors make $f_ j$ converge locally uniformly to some $f\in \mathcal O(B)$. For a fixed $j$, every later approximation controls its error on $C_ j$, since $C_ j\subset K_{m-1}$ for $m>j$. Consequently
\[ \sup _{C_ j}|f-g_ j| \leq \sum _{m\geq j}\epsilon _ m \leq 2^{-j-1}\delta _ j<\delta _ j. \]
Returning to the original notation gives the stated tolerances.
$\square$
Pome
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