Lecture 5: Some Basic Constructions in Symplectic Topology

In complex geometry the blowing-up operation amounts to replace a point in a space by the space of complex tangent lines through that point. It is a local operation which can be explicitly written down as follows. Consider the blow-up of Cn at the origin. This is the complex submanifold of Cn × CPn−1 C̃ ≡ {((z1, z2, · · · , zn), [w1, w2, · · · , wn])|(z1, z2, · · · , zn) ∈ [w1, w2, · · · , wn]}. Note that as a set C̃n = Cn \ {0} t CPn−1, with CPn−1 being the space of complex tangent lines through the origin. CPn−1 ⊂ C̃n is called the exceptional divisor. The projection onto the first factor Cn induces a biholomorphism between C̃n \CPn−1 and Cn \ {0}, and collapses the exceptional divisor CPn−1 onto the origin 0 ∈ Cn. On the other hand, the projection onto the second factor CPn−1 defines C̃n as a holomorphic line bundle over CPn−1. Blowing up is often used in resolving singularities of complex subvarieties. We give some examples next to illustrate this.