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I want to prove the uniqueness and existence of integral curve on a complex manifold, that is, find a curve $c$, parametrized by a small neighborhood $\Delta \subset \mathbb{C}$ with $c(0)=p$ in a complex manifold $M$, such that $\frac{\partial c}{\partial t}=V(c(t))$, $V$ is a given vector field (a local section of holomorphic tangent bundle )on $M$.

If we see $M$ as a smooth manifold, we have classic result of the uniqueness and existence of real integral curve, by theorems in ODE theory. If we solve $\frac{\partial c}{\partial t}=v(c(t))$ by splitting real and imaginary part and use theorem of uniqueness and existence of solution, we should also get a smooth solution of the ODE, but is this solution holomorphic, i.e. in locally coordinates, $c$ varies holomorphically with $t$?

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  • $\begingroup$ Does it help to notice that $\dfrac{\partial c}{\partial\bar t}$ satisfies the differential equation $$\frac{\partial}{\partial t}\left(\frac{\partial c}{\partial\bar t}\right) = \frac{\partial V}{\partial z}(c(t))\frac{\partial c}{\partial\bar t}?$$ $\endgroup$
    – Ted Shifrin
    Sep 27 at 17:40
  • $\begingroup$ Is $V$ assumed to be a holomorphic section? (Not every smooth section of the holomorphic tangent bundle is holomorphic, in the same way not every smooth, complex-valued function is holomorphic.) $\endgroup$
    – Andrew D. Hwang
    Sep 27 at 18:43
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    $\begingroup$ @AndrewD.Hwang V is just a local section of holomorphic tangent bundle, not a holomorphic section. $\endgroup$
    – user24918
    Sep 27 at 18:47
  • $\begingroup$ @TedShifrin I see that$\frac{\partial c}{\partial \bar{t}}=0$ is solution of this differential equation, in this case, $c$ is holomorphic, I think this differential equation has other solutions, so there is a holomorphic solution and possibly other solutions that $c$ is not holomorphic? $\endgroup$
    – user24918
    Sep 27 at 18:54
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    $\begingroup$ Oh, I was assuming a holomorphic vector field. Without that hypothesis, you can’t expect the local flow to be holomorphic. $\endgroup$
    – Ted Shifrin
    Sep 27 at 19:20

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