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?
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$\begingroup$ Just follow the usual proofs from the real case. $\endgroup$ – Ben McKay Feb 4 '19 at 20:16
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$\begingroup$ @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? $\endgroup$ – Flavius Aetius Feb 5 '19 at 16:44
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$\begingroup$ @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? $\endgroup$ – Flavius Aetius Feb 5 '19 at 16:46
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$\begingroup$ @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. $\endgroup$ – Flavius Aetius Feb 5 '19 at 18:29
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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.
There is also a global version of this theorem which says the following:
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.
