McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic form, then in the symplectic blow-up (with blow-down map $f$), the new symplectic form is $f^{\ast}\omega+\epsilon\alpha$, for $\epsilon$ small enough, and where $\alpha$ is a particular $2$-form (satisfying certain conditions) on the canonical line bundle over the projectivization of the normal bundle of the submanifold.

If one instead has a complex, compact manifold that admits a symplectic structure, then is it known how the holomorphic/complex blow-up procedure along a (symplectic) submanifold affects the ambient symplectic structure, if at all?


In the case when you are blowing up a smooth submanifold this actually coincides with the symplectic blow-up (with the symplectic form forgotten). For simplicity, let's take the submanifold to be a point, because the general case is not more difficult. Now symplectic blow-up replaces a standard symplectic ball $B_\varepsilon\subset M$ of that point with the unit disc bundle of the tautological line bundle over $\mathbb{CP}^{n-1}$. The general case is just a fibered version of this description. From this it's easy to see that topologically it coincides with the blow up of a point defined using complex coordinates.

The only difference is that when the Kahler manifold $M$ is viewed as a symplectic manifold, you need to equip the manifold $\mathit{Bl}_p(M)$ you have just obtained with a symplectic form. This additional piece of geometric structure can be set to be $\pi^\ast\omega_M+\varepsilon\omega_{FS}$, which is exactly the one that you have written down. Locally, this is nothing else but the usual symplectic structure on the total space of the tautological line bundle over $\mathbb{CP}^{n-1}$ or a projective bundle over the center that you want to blow up in the general case, where by $\omega_{FS}$ we mean a compactly supported 2-form which restricts to the Fubini-Study metric on the exceptional divisor in the case of blowing up a point and a symplectic form which fiberwisely coincides with the Fubini-Study metric when restricting to the exceptional divisor in the general case. This is the 2-form $\alpha$ appeared in your description of a symplectic blow-up.

Generalizing this, one can also define the notion of a symplectic flip, which is done by combining the above idea with the principle of Kempf-Ness. Namely we start with some symplectic manifold $M_+$, and cover it by two open symplectic submanifolds $W_+$ and $U$. When performing a symplectic flip, the symplectic structure on $U$ us unchanged. To make use of the variation of GIT construction, we require the exsistence of another symplectic manifold $Y$ which carries a Hamiltonian circle action, whose moment map is denoted by $\mu$, such that $W_+=\mu^{-1}(1)/S^1$. Passing to the symplectic reduction at the level $-1$, we get another open symplectic manifold $W_-$. Now the symplectic manifold $M_-$ obtained by applying the symplectic flip $M_+\dashrightarrow M_-$ is simply defined by replacing $W_+$ with $W_-$. In the case when $M_+$ is an algebraic manifold, one can check that it coincides with the usual definition of a flip, by forgetting the symplectic structure on $M_-$.


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