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Eliashberg & Gromov sketched a proof in their paper "Convex symplectic manifolds" (Section 3.4). Written in the 4-dimensional case it says: For $r>0$ define the subspace $X(r)\subset\mathbb{R}^4$ to …

Yes, given two contact 3-manifolds $(M_1,\xi_1)$ and $(M_2,\xi_2)$ we can form their contact sum $(M_1\# M_2,\xi_1\# \xi_2)$ and then ECH decomposes as the tensor product of the corresponding ECH's of …

I assume here that $\lambda$ is degenerate but not too badly, i.e. such that its orbits are smooth manifolds (possibly with boundary) and $d\lambda$ has constant rank along them (along with another mi …

Why not look at it in specific examples? Say the height function $H$ on the sphere $S^2\subset\mathbb{R}^3$. Or how it doesn't work for the obvious rotations on the torus. The canonical context in wh …

For your $n=2k$, $\mathbb{C} P^n\#\mathbb{C} P^n$ does not even admit an almost complex structure, so it cannot be symplectic. See also: 1) Goertsches-Konstantis' paper "Almost complex structures on …

Assume your 4-manifold is minimal (otherwise there are multiply covered exceptional spheres which potentially give noncompactness). Then it's a computational check to see that: If $d(\alpha)=0$ with $ …

The key use of SW theory was to show that $Gr(e)=Gr(c_1(K)-e)$. For the moment, take this equality as granted. If $Gr(e)\ne0$ then there must exist a $J$-holomorphic curve $C\to X$ such that $[C]=e$, …

I was there during this IHES conference to scribe the lectures and also give a discussion session on it. So in case it helps, my notes from both of these are available here. Your thought on the reaso …

Without elaborating much there are three key points, with the first two laying the bedrock for the third: ECH counts J-curves without caring about most information of the actual branched covers of su …

Moral: This is giving a glimpse at the troubles with branched multiply-covered $J$-holomorphic curves. The virtual-dimension is a Fredholm index, involving $\chi(C)$ of your curve. If $C$ is multiply- …

This is intended to compliment Robert Bryant's post, and answers Ritwik's question in his comment. For $\dim_\mathbb{R} N>2$ this is not a useful notion. Indeed, for a generic $(N,J_N)$ there are no …

The "2-dimensional" aspect concerns the 2-dimensional curves and their punctures/nodes, and this went into the SFT Hamiltonian/potential. SFT contains the GW invariant (viewing a closed manifold as a …

So the key is that we are taking the standard complex structure on $\mathbb{C}^n$ (hence the ball in it). Then the pseudo-holomorphic preimage $C$ is actually a holomorphic curve, and such surfaces ar …

Spectral flow is the standard way a sign is associated to a point in a zero-dimensional moduli space of curves (I am not sure what you mean by VFC here, it's 0-dimensional). This involves orienting th …

Exercise: The non-symplectic $K3\#n(S^1\times S^3)$ for $n>0$ with the spin-c structure induced from the canonical one on $K3$; it will have $n$-dimensional SW moduli. If you want symplectic examples: …