Do holomorphic symplectic manifolds admit (high codimension) embeddings in some standard space? Per the Whitney embedding theorem, any manifold $M$ can be embedded into a sufficiently high dimensional Euclidean space.
According to Gromov's h-principle for contact embeddings, any contact manifold admits high codimension embeddings contact embeddings into the standard contact $\mathbb{R}^{2n+1}$, so in particular any exact symplectic manifold can be thusly embedded.  It is also known that integral symplectic manifold can be embedded into a sufficiently high dimensional projective space.
Meanwhile in complex geometry, certainly not all complex manifolds embed into $\mathbb{C}^N$ for any $N$, or for that matter (as far as I know) into any particular space.  But large classes do; the Stein manifolds into $\mathbb{C}^N$, and the integral Kahler manifolds into projective space.

Which classes of holomorphic symplectic manifolds admit (high codimension) embeddings as holomorphic symplectic submanifolds of some standard holomorphic symplectic space?

I would particularly like to know:

When can a holomorphic symplectic manifold $W$ be embedded as a (high codimension) holomorphic symplectic submanifold of the holomorphic contact manifold $\mathbb{P} T^* M$ for some complex manifold $M$?

By a symplectic submanifold of a contact manifold, I mean that for some choice of the contact form $\lambda$, the restriction of $d \lambda$ gives the symplectic form.
For this question I allow $M$ to be arbitrary (e.g. you may take $M = W$), so the complex geometry itself should give no obstructions.
I am particularly interested in complex symplectic manifolds which are conic in the sense of admitting a $\mathbb{C}^*$ action which scales the symplectic form.  These are complex analogues of the exact symplectic manifolds mentioned above, since a $\mathbb{R}$ action scaling the symplectic form is a vector field $Z$ with $\omega = Z \omega = d i_Z \omega + i_Z d \omega = d \omega (Z, \cdot)$.
 A: You can ask a similar question for
holomorphic contact manifolds: when
such a manifold can be embedded to
a projectivization of the cotangent bundle.
The answer is known (for projective holomorphically
contact manifolds), see for example the survey
https://arxiv.org/abs/1805.08548
Theorem 5.3 (Kebekus, Peternell, Sommese, Wiśniewski and Demailly)
If $(X, F)$ is a projective complex contact manifold,
then $(X, F)$ is either

*

*the projectivisation of the cotangent bundle of a projective manifold


*a projective space


*a contact Fano manifold such that $Pic X = {\mathbb Z} [L]$ (that
is, all the complex line bundles on X are
isomorphic to tensor powers of $L:= TX/F$ or its dual).

It is clear that a projective contact manifold with $b_2=1$
cannot have a contact embedding to ${\mathbb P}T^*M$,
because a contact submanifold in ${\mathbb P}T^*M$ has
$b_2 \geq 2$; the converse follows from this theorem.
Your question about holomorphically
symplectic manifolds with $C*$ -action is more
or less reduced to this one, if you are interested in
$C*$ -equivariant embeddings. Without
equivariance, it is more tricky, but I expect
that the answer for a holomorphic contact
manifold with $b_2=1$ is still negative,
except $CP^n$.
All the best
Misha
