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Suppose $f_1$ and $f_2$ are two commuting symplectomorphisms of the sphere $\mathbb S^2$, of orders different from $2$. Is it possible to deform the pair $(f_1,f_2)$ to the pair of identity maps via a …

Here is one example - take an elliptic curve without an automorphism of order $4$ with a fixed point. Note at the same time that any torus $T^2$ with an area form has an area preserving automorphism o …

Suppose that $(M,\omega)$ is a (connected compact) symplectic manifold with a Hamiltonian $S^1$-action given by Hamiltonian $H$. I would like to find a reference for the fact that every level set of …

Let $M^{2n}$ be a symplectic manifold and let $M^{2n-2}$ be a symplectic submanifold. How to construct a non-trivial Hamiltonian $S^1$-action on $M^{2n-2}$ on a small neighborhood of $M^{2n-2}$, that …

That's true and follows from the fact a vector space with a countable dense subset can't have an uncountable number of open subsets that don't intersect pairwise. Here the vector space is the space of …

It looks indeed that this question is not as classical is it sounds, so let me provide a 2010 reference to a more general statement, at least to show that there is a reference. This is Proposition 3. …

I am aware that the following result is a classical one (by now). But I am not able to understand who proved it. What should be a proper reference to this statement? Theorem. Let $M^4$ be a compact s …

Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $v$ …

I would like to find some good references (or any insight) that would help me to understand a few articles mentioning rigidity of Hirzeburch genus. One of the consequences of this phenomenon is that w …

I would like to know if following is correct. Statement. Suppose we have a smooth (i.e., $C^\infty$) almost complex structure on $\mathbb R^4$ and $C_1, C_2$ are two $J$-holomorphic curves passing t …

It turns out that it is nice to read books. The answer to the weaker version of my question with $O|z|^{n+1}$ term is contained on page 17 of McDuff-Salamon book [MS] (no need of Micallef-White!): htt …

I am certain that the following result holds, but was not able to find a reference. Do you know one? Or maybe you can give a short proof? Statement. Let $(M^4,\omega)$ be a compact symlectic manifold …

It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider mirr …

Is there an example of a smooth vector field $v$ on $S^3$ such that $v$ preserves a volume form and $v$ is not a Reeb vector field? Recall that $v$ is a Reeb vector field if there exists a contact $ …

Question 1. Can a reflexive Delzant polytope of some dimension contain a $2$-face with more than $11$ edges? Motivation. I would like more generally to get an answer to the following question: Questio …