Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 17 / 4 / b

True. Use the normalization in which the blowup weight $\lambda$ is the area of an exceptional projective line, equivalently the capacity $\pi r^2$ of the ball removed. The auxiliary volume formula for a point symplectic blowup in real dimension $2n$ isIt applies to each successive blowup, irrespective of where the point is chosen. Geometrically the removed standard ball has volume ${\pi }^n{r}^{2n}/n!$, while its replacement divisor has zero $2n$-dimensional volume. If $V$ is the original symplectic volume, after $k$ equal-weight blowups the volume is $V-k{\lambda }^n/n!$. A nonempty closed symplectic manifold has strictly positive volume, soThis finite upper bound suffices; it does not assert that every number below the bound is realizable. A convention taking the weight to be the radius instead changes the fixed decrement to ${\pi }^n{\lambda }^{2n}/n!$, with the same conclusion.