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I will denote by $\mathbb{C}^*$ the punctured complex plane, $\mathbb{C} \setminus \{0\}$. Let's say I have a holomorphic map on the punctured plane, $w: \mathbb{C}^* \to \mathbb{C}$, such that the map $e^{w}: \mathbb{C}^* \to \mathbb{C}^*$ has a (holomorphic) antiderivative $F: \mathbb{C}^* \to \mathbb{C}$.

Obviously I have a homotopy between $e^{w}$ and the constant map 1 where the intermediate maps are holomorphic functions $\mathbb{C}^* \to \mathbb{C}^*$: i.e., I simply consider $e^{tw}, t \in [0,1]$. However, the intermediate maps of this homotopy do not necessarily have antiderivatives, obviously. I was wondering if this homotopy could be modified in some way to ensure that the intermediate maps $\mathbb{C}^* \to \mathbb{C}^*$ do have antiderivatives.

Is there a theorem that gives this property, or if not, does anyone have any ideas as to how I could approach this problem? (I'd appreciate any ideas for approaches - even if they are vague. Or any references that might be helpful. Thank you!)

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  • $\begingroup$ I think that some Atiyah-Singer type business might be the appropriate route here. Indeed, if we can solve the equation $(\partial_t-w)\partial_zF=0$ for some $F:\mathbb{C}^* \times \mathbb{R} \to \mathbb{C}$, then we'll be done. As the relevant differential operator here is elliptic, but the domain of the operator is topologically nontrivial, the natural theorems to be invoking are the index theorems. $\endgroup$ – Or Eisenberg Dec 27 '18 at 18:46

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