Do holomorphic symplectic manifolds admit (high codimension) embeddings in some standard space?

Question

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 $\omega = Z \omega = d i_Z \omega + i_Z d \omega = d \omega (Z, \cdot)$.

Answer 1

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).

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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