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There are some very good reasons why the majority of calculations are done for algebraic manifolds. Maybe the most naive reason is as follows: it is harder to solve PDEs than to draw lines through tw …

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 …

These two symplectic manifolds are canonically symplectomorphic. Notice first, that the map $\mu$ vanishes on the sub-bundle of $T^* M$ of 1-forms vanishing on the fibers of the fibration $M\to X$. …

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 …

This is not a complete answer to the question, and I don't know if a complete answer is written down anywhere in the literature. In the first revison of the answer I tried to adress all the 10 commen …

As far as I know your question is completely open. At the present moment no one knows if for $n>2$ there is a symplectic structure on any manifold homotopic to $\mathbb CP^n$ but not diffeomorphic to …

Gromov Witten invariant is supposed to "count" the number of curves but it can happen that the dimension of the space of curves that you want to count is larger than you expect. For example on a 3-dim …

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 …

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 …

Since on any symplectic manifold you can easily find an almost-Kahler metric, this question can be reformulated as follows: "In the realm of symplectic geometry , to what extent , the hard Lefschetz …

Let me say first, that I really like this question. Very unusual question about such well known things (in fact I did not know even that the real quintic $\sum_i x_i^5=0$ is $\mathbb RP^3$). This is …

This answer is rewritten and include more details First of all I highly recommend you the article of Paul Biran From Symplectic Packing to Algebraic Geometry and Back available on the page http://ww …

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 …

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 …

I think, that in order to answer this question it is worth to conisder the complex algebraic analog of this question. Namely, suppose we have a $\mathbb C^*$ action on a projective manifold $V^n$. Can …