Holomorphic version of Darboux's theorem

I would like to ask if there is a holomorphic version of Darboux's theorem. More concretely, given a holomorphic symplectic manifold $$(X, \omega)$$ is there a local holomorphic symplectomorphism from $$(X, \omega)$$ to $$(\mathbb{C}^{2n}, \omega_0)$$ where $$\omega_0$$ is the holomorphic equivalent of the standard symplectic form in $$\mathbb C^{2n}$$. To put it differently, is it true that $$X$$ locally looks like a cotangent bundle? Do you have a reference?

• Just follow the usual proofs from the real case. – Ben McKay Feb 4 '19 at 20:16
• @Aknazar Kazhymurat: Thank you for the link. It is an interesting paper. They mention the "holomorphic Darboux theorem" which is what I am looking for. But the theorem itself is not stated. I guess that is because it is the exact holomorphic version of the classical Darboux theorem? – Flavius Aetius Feb 5 '19 at 16:44
• @Ben McKay: So there is a holomorphic Dardoux's theorem and it is exactly as the real one except that we replace "smooth" by "holomorphic". Is that right? No hidden traps there? Do you have a standard reference? – Flavius Aetius Feb 5 '19 at 16:46
• @AknazarKazhymurat: I mean "exact" in the sense of "precisely the same". Yes, your right, there is a formulation of that theorem in one of the proofs of another theorem. – Flavius Aetius Feb 5 '19 at 18:29

Suppose $$Y\subset X$$ is a complex submanifold and we're given two holomorphic symplectic forms $$\omega_0$$ and $$\omega_1$$ on (a neighbourhood of $$Y$$ in) $$X$$. Then I will prove that there exist two open neighbourhoods of $$Y$$ in $$X$$ and a biholomorphism $$\varphi$$ between them s.t. $$\varphi^*\omega_1 = \omega_0$$.
Define $$\omega_t = (1-t)\omega_0 + t\omega_1 = \omega_0 + \sigma$$. We are looking for a smooth family of holomorphic maps $$\varphi_t$$ s.t. $$\varphi^*_t\omega_t = \omega_0$$, $$\omega_0 = id$$. By the standard Moser trick we're reduced to finding a family of holomorphic vector fields s.t. $$\sigma = -d\iota_{\eta_t}\omega_t$$ as the flow of a holomorphic vector field is holomorphic.
Find some $$\alpha$$ s.t. $$\sigma = d\alpha$$ (it is easy to construct it explicitly by introducing some smooth deformation retraction of a neighbourhood of $$Y$$ in $$X$$ to $$Y$$). Decompose $$\alpha = \alpha^{1,0} + \alpha^{0,1}$$ in such a way that $$\alpha^{1,0}\in \Lambda^{1,0} X$$, $$\alpha^{0,1} \in \Lambda^{0,1}X$$. Then $$\partial \alpha^{1,0} = \sigma$$, $$\overline{\partial} \alpha^{1,0} = -\partial \alpha^{0,1} = \gamma$$, $$\overline{\partial}\alpha^{1,0} = 0$$. Now $$\gamma$$ is $$\partial$$ and $$\overline{\partial}$$-exact hence local $$dd^c$$-lemma* can be applied to it and we can find a function $$\rho$$ s.t. $$\gamma = -\partial\overline{\partial}\rho$$. Denote by $$\beta$$ the form $$\alpha^{1,0} - \partial \rho$$. Then $$\partial\beta = \partial\alpha^{1,0} = \sigma,$$ $$\overline{\partial}\beta= 0.$$ We've reduced the problem to solving the equation $$\beta = - \iota_{\eta_t}\omega_t$$ which has a unique holomorphic solution.
Let $$X$$ be a compact complex manifold equipped with two holomorphic symplectic forms in the same cohomology class. Then there exists a holomorphic automorphism $$\varphi$$ of $$X$$ that pullbacks one form to another. In the proof we use that these forms can be connected by a smooth path of holomorphic symplectic forms. It relies on Yau's theorem as we need to use the global $$dd^c$$-lemma hence need a Kähler structure.
*In fact it is not obvious how to use $$dd^c$$-lemma when $$Y$$ is not a point. I conjecture we need to require that $$dd^c$$-lemma holds on $$Y$$ which is automatically true when $$X$$ is hyperkähler, in particular, when $$X$$ is compact (by Yau's theorem) but I need to think about it.