Symplectic blowup ($\tilde{M}$)

A symplectic blowup at a point removes a small symplectic ball and collapses its Hopf-circle boundary fibers, replacing the center by an exceptional divisor $E\cong {\mathbb{CP}}^{n-1}$. The resulting form agrees with the original form away from the surgery region, and its integral over a projective line in $E$ records the blowup size. In real dimension four, $E\cong {\mathbb{CP}}^1$ and $E^2=-1$.

For a symplectic blowup of a $2n$-dimensional symplectic manifold, the displayed formula gives its First Chern class. The exceptional divisor has normal bundle $\mathcal{O}(-1)$; the discrepancy coefficient is $n-1$. In dimension four it is one.

For a closed $2n$-dimensional symplectic manifold, a symplectic blowup of size $\epsilon =\pi R^2$ removes the symplectic volume of the radius-$R$ ball. Its exceptional divisor has lower dimension and contributes no ambient volume. The displayed volume loss shows that different size choices can give forms that are not related by a symplectomorphism.