# Integral curve on complex manifold

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 $$t$$?

• Does it help to notice that $\dfrac{\partial c}{\partial\bar t}$ satisfies the differential equation $$\frac{\partial}{\partial t}\left(\frac{\partial c}{\partial\bar t}\right) = \frac{\partial V}{\partial z}(c(t))\frac{\partial c}{\partial\bar t}?$$ Sep 27 at 17:40
• 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.) Sep 27 at 18:43
• @AndrewD.Hwang V is just a local section of holomorphic tangent bundle, not a holomorphic section. Sep 27 at 18:47
• @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? Sep 27 at 18:54
• Oh, I was assuming a holomorphic vector field. Without that hypothesis, you can’t expect the local flow to be holomorphic. Sep 27 at 19:20