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

• This is not what you are asking for, but you might still be interested by this preprint which shows that a compact holomorphic symplectic manifold cannot be embedded as a Poisson submanifold of a projective space (Corollary 1.8).
– abx
Oct 13 at 19:44

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