Holomorphic version of Darboux's theorem

Question

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?

Answer 1

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.

Answer 2

To put it differently, is it true that X locally looks like a cotangent bundle?

This is false. Indeed, take an elliptic curve $C$ inside an elliptic K3 surface. If it had a neighbourhood $U$ which is biholomorphic to a cotangent bundle to an elliptic curve, we would have (after shrinking $U$ if necessary) $U=C \times \Delta$ where $\Delta$ is a disk. This easily follows because the cotangent bundle to an elliptic curve is trivial.

However, existence of such a neighbourhood would imply that this family of elliptic curves is locally trivial ("isotrivial", as people usually say). It is very easy to find an elliptic K3 surface which is not isotrivial, see for example https://arxiv.org/abs/1406.1233 .

holomorphic version of Darboux's theorem

The theorem is more or less trivial, if you need just to have Darboux coordinates. Follow the standard argument with the Moser's lemma, it works the same way in the holomorphic context. Just in case, here is the reference for holomorphic Moser lemma, if you need it (in a more general context indicated by Anna's answer to this question): https://arxiv.org/abs/2109.00935