TITLE: Symplectic blow-up
QUESTION [5 upvotes]: Blow-ups of points can also be performed in the symplectic category; for a given point $p\in (X,\omega)$ we choose a Darboux chart around $p$ and then use the symplectic cut corresponding to the standard hamiltonian S^1-action (Diagonal action) to define the blow-up.
It should be possible to define blow-ups of general symplectic submanifolds (which is J-holomrphic with respect to some fixed compatible almost complex structure) in a similar way.
Given $X\subset Y$ (a symplectic and J-holomorphic submanifold) we need to put a symplectic structure on a neighborhood of $X$ in $N_X^Y$ (normal bundle) which by symplectic neighborhood theorem is symplectomorphic to a neighborhood of $X \subset Y$ and such that the action of $S^1$ (multiplication by $e^{i\theta}$) is Hamiltonian.
I can't find a reference in the literature where the details are written? Is it any where out there?

REPLY [3 votes]: I think "Birational equivalence in the symplectic category" - Invent.Math (by V.Guillemin and S.Sternberg) covers all details that you want. (Section 8).