Constructing a homotopy of nonzero holomorphic functions using local homotopies

I'll denote by $$\mathbb{C}^*$$ the punctured complex plane $$\mathbb{C} \setminus \{0\}$$. Say that I've got some open cover $$\{V_j\}_{j \in J}$$ of the closed unit interval $$[0,1]$$, and continuous functions $$w_j: V_j \times \mathbb{C}^* \to \mathbb{C}^*$$ such that, for any $$t \in V_j$$,

(a) $$w_j(t, \cdot)$$ is a holomorphic function $$\mathbb{C}^* \to \mathbb{C}^*$$ and

(b) $$w_j(t, \cdot)$$ has a holomorphic antiderivative, say $$F: \mathbb{C}^* \to \mathbb{C}$$.

I want to find a function $$u: [0,1] \times \mathbb{C}^* \to \mathbb{C}^*$$ that such that, for fixed $$t \in [0,1]$$, $$u(t, \cdot)$$ is still holomorphic and still has a holomorphic antiderivative.

My first thought was to use partitions of unity with respect to this cover, say $$\{p_j\}_{j \in J}$$ and to set $$u(t, \cdot) = \sum_{j} p_j(t)w_j(t, \cdot)$$. Of course, such a $$u$$ does still have an antiderivative for each $$t$$ but now we can't be sure whether the image of $$u$$ still lies in $$\mathbb{C}^*$$.

I get that this is a bit vague, but I'd appreciate any good advice and/or ideas...can someone think of another way of constructing $$u$$. Or perhaps I can pursue the partitions of unity idea, provided $$w_j$$ and/or the open cover have some additional features, and if so, what kinds of features would enable this? I'd appreciate any ideas.

• You don't seem to require any compatibility relations between this $u$ and the previous data, so it's unclear to me what kind of $u$ do you want to construct. Could you give some additional precisions ? – Nicolas Hemelsoet Dec 20 '18 at 11:57
• P.S I think your previous idea should work with some additional modifications, for example you can assume that $J$ is countable and fix an arbitrary ordering on it. Now it should be possible to inductively build the $p_j$ so that the function is still $\Bbb C^*$ valued. – Nicolas Hemelsoet Dec 20 '18 at 11:59
• Thanks for your remarks! Is it actually possible to build the partition of unity inductively so that the function is still nonzero? I'm not quite sure if I understand what you mean! – Acton Dec 21 '18 at 2:00