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This can always be done. Let's first treat the case when $\Sigma$ is not a torus. Then take any symplectic $4$-manifold $(M,\omega)$ where $\Sigma$ can be embedded as a Lagrangian surface. Now, take a …

I had a look at the paper of Givental, https://math.berkeley.edu/~giventh/papers/tor.pdf and don't see this statement... If this statement were true, Conjecture 6.1 of Eliashberg from 2015 would be …

I hope that the following answers some parts of the question. 1) a) It is not true that a generic contactomorphim doesn't have fixed points. For example, let $M$ be the three-dimensional torus that …

Nick Lindsay and have just proved that such a manifold indeed exists. And surprise, surprise, this is Tolman's manifold. See Theorem 1.3 and Corollary 1.4 of our paper: https://arxiv.org/abs/1912.027 …

I'll give a positive answer for two generic totally real planes in $\mathbb C^2$. I believe this generalises to larger $n$, though I don't prove it - just give a possible plan of a proof with one step …

A positive answer to this question should follow from Berger's holonomy classification and the following statement, which I believe is correct: Statement. For any dimension $n$ there exists a positi …

One way to prove this is as follows. First, from the assumption $B(1)\subset \mathbb C^2$ and the centre of $B(1)$ is $(0,0)$. Now we need the following two claims. Claim 1. Any straight geodesic u …

This doesn't need to hold. For example, if one takes a $(T^4,\omega)$ with a constant symplectic structure $\omega$, in order for it to have a fibration by Lagrangian tori one should be able to find a …

Concerning 2) one can, of course, take the product of two curves of higher genus to get a counter-example. In general, to have a statement as you want, one should look for rigid complex surfaces of ge …

Here is an answer to the REFINED question given to me by Richard Thomas. In this refined version one looks for an example such that the cohomology classes of two symplectic forms coincide. In a late …

I have to apologize, in fact the answer to the second question is still unknown. Namely, up to now all known symplectic manifolds of dimension 4 that have negative Euler characteristic are blow ups of …

Let us consider the case of toric varieties of real dimension $4$ and prove they can not be represented as such a quotient unless they have second Betti number $1$ or $2$. Proof. Let us introduce …

There are counter-examples, hope they answer your question completely, just take any non-simply connected $G$ and consider its action on $T^*G$. The simplest case is: Let $M$ be the cylinder $S^1\tim …

In a certain sense, symplectic geometry (or safer to say symplectic topology) as we know it now was not existing before Arnold formulated these conjectures. So many would say that Arnold conjectures g …

I think the answer to your question should be positive and below is a sketch of what should work (I think). Any real analytic manifold can be realised as the real part of a complex projective manifol …