# Holomorphic Weinstein Lagrangian neighborhood theorem

The Weinstein Lagrangian neighborhood theorem says that if $$(M,\omega)$$ is a symplectic manifold and $$L\subset M$$ is a Lagrangian submanifold, then there are neighbourhoods $$U$$ of $$L$$ in $$M$$, and $$U'$$ of the zero-section in $$T^*L$$, and a symplectomorphism $$U\to U'$$ which restricts to the identity on $$L$$.

Question: Is there a holomorphic version of this theorem for holomorphic-symplectic manifolds and complex-Lagrangian submanifolds?

Presumably, we would need some additional conditions on the embedding since the tubular neighborhood theorem doesn't necessarily hold in the holomorphic setting.

A classic example that shows that WLNT doesn't always hold in the holomorphic category is an elliptically fibered $$K3$$ surface.
A K3 surface $$S$$ is a compact complex symplectic manifold of complex dimension $$2$$. Any smooth curve $$C\subset S$$ is a Lagrangian submanifold. If the Darboux Theorem were true in the sense that a neighborhood of the curve were always symplectically biholomorphic with a neighborhood of the zero section of the cotangent bundle, then, for a nonsingular elliptic curve $$C\subset S$$, the neighborhood would be a product and so the $$1$$-parameter family of nearby deformations of the curve $$C$$ in $$S$$ would all be isomorphic to it, i.e., they would all have the same $$j$$-invariant. However, this is known not to be the case: When you have an elliptically fibered $$K3$$, the $$j$$-invariant of the elliptic fibers is not constant.
If one doesn't need the ambient symplectic manifold to be compact, there is an easier example: Let $$z$$ be the standard complex coordinate on $$\mathbb{C}$$ and let $$w$$ be the standard complex coordinate on the upper half-plane $$\mathbb{U}^+\subset\mathbb{C}$$. Let $$\mathbb{Z}^2$$ act (freely) on $$X = \mathbb{C}\times\mathbb{U}^+$$ by $$(m,n)\cdot(z,w) = (z + m + nw,\,w).$$ Let $$M$$ be the quotient of $$\mathbb{C}\times\mathbb{U}^+$$ by this $$\mathbb{Z}^2$$-action. This action preserves the symplectic form $$\mathrm{d}z\wedge\mathrm{d}w$$, so that $$M$$ is a symplectic surface, as well as the projection $$w:M\to \mathbb{U}^+$$, whose fiber in $$M$$ over $$a\in \mathbb{U}^+$$ is a torus $$w^{-1}(a) \simeq \mathbb{C}/(\mathbb{Z}{+}\mathbb{Z}a)$$. These tori are Lagrangian curves in $$M$$, but no neighborhood of such a torus is biholomorphic to the product of the torus and a disk, precisely because the tori near a given $$w$$-fiber (which are all $$w$$-fibers since $$w$$ would have to be constant on any compact, connected curve in $$M$$) are not biholomorphic to the given $$w$$-fiber.
Added Remark: However, this result is particular for the zero section of the cotangent bundle of $$L$$. It is not in general true that, for any two holomorphic symplectic structures $$\omega_1$$ and $$\omega_2$$ on a complex manifold $$M$$ that have a common holomorphic Lagrangian submanifold $$L\subset M$$, there exist open $$L$$-neighborhoods $$U_1$$ and $$U_2$$ in $$M$$ such that there exists a biholomorphic mapping $$\phi:U_1\to U_2$$ fixing $$L$$ and satisfying $$\phi^*\omega_2 = \omega_1$$.
For example, taking the above torus fibration $$w: M\to \mathbb{U}^+$$, for different nonvanishing holomorphic functions $$f_1,f_2:\mathbb{U}^+\to\mathbb{C}$$, the symplectic forms $$\omega_i = f_i(w)\,\mathrm{d}z\wedge\mathrm{d}w$$ for $$i=1,2$$ will not generally be locally holomorphically symplectomorphic near any $$w$$-fiber.