Search Results

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 …

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 …

$\mathbb CP^2\#\overline{K3}$ (trivial SW because it is a connected sum with $b^2_+>0$ for both pieces). This is an example without having to use the Bauer-Furuta invariants (contrast with Kyle's comm …

Michael Hutchings' lecture notes were precisely for this purpose; posted on his webpage: https://math.berkeley.edu/~hutching/ Lecture Notes on Morse Homology (With an Eye Towards Floer Theory and Pseu …

This might suffice for you, it is not published and only slightly longer than Etnyre's sketch, but without exercises. This has been shown in the PhD thesis of Noah Daniel Goodman (a student of Etnyre) …

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 $[C]=e$, …

Convergence to cusp curves is the original compactification by Gromov, whereas convergence to stable maps is the compactification by Kontsevich (a cusp curve corresponds to the image of a stable map). …

Ozsvath and Szabo constructed HF as a topological interpretation of SWF, and they noticed many links between the two. Roughly speaking, their Euler characteristics are the same, and the analog of the …

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 …

Just to finalize comments since people are upvoting the question: The canonical reference is Taubes' "Casson's invariant and gauge theory" which makes the statement rigorous and has all the definition …

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 …

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: …

This is not Taubes' moduli space. Taubes has many more constraints on the currents, so that his "moduli space" is really a finite set of points (for generic $J$), and he requires special weightings on …

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 …

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 …