Let $X$ be a subvariety in $M$. "Deformation to the normal cone" is a holomorphic deformation of a neighbourhood of $X$ in $M$ over the disk such that its central fiber is the total space of the normal bundle $NX$, and the rest of the fibers are $M$.
I want to have a holomorphically symplectic version of this statement. The following conjecture seems to be true; I think I have a proof (ugly), but I am afraid it's already known.
Let $(M, \Omega)$ be a holomorphically symplectic manifold, and $X\subset M$ a compact holomorphic Lagrangian subvariety. Assume that $M$ admits a proper, birational map which contracts $X$. Then there exists a smooth, holomorphic deformation of a neighbourhood of $X$ in $M$ over the disk $\Delta$, such that its central fiber is biholomorphic to a neighbourhood of $X$ in $T^*X$, the rest of the fibers are biholomorphic to a neighbourhood of $X$ in $M$, and the holomorphic symplectic form on $T^*X$ can be smoothly extended to the holomorphic symplectic form on the rest of the fibers.
This statement is provably false for some other holomorphic Lagrangian subvarieties, such as a smooth fiber in a sufficiently generic proper holomorphically Lagrangian fibration. I know there are MO users who know the subject better than me, any help will be highly appreciated.
