Blowing Up - Blowing Up Submanifolds in Complex Manifolds

Blowing Up Submanifolds in Complex Manifolds

More generally, one can blow up any codimension-k complex submanifold Z of Cn. Suppose that Z is the locus of the equations, and let be homogeneous coordinates on Pk - 1. Then the blow-up is the locus of the equations for all i and j, in the space Cn × Pk - 1.

More generally still, one can blow up any submanifold of any complex manifold X by applying this construction locally. The effect is, as before, to replace the blow-up locus Z with the exceptional divisor E. In other words, the blow-up map

is a birational mapping which, away from E, induces an isomorphism, and, on E, a locally trivial fibration with fiber Pk - 1. Indeed, the restriction is naturally seen as the projectivization of the normal bundle of Z in X.

Since E is a smooth divisor, its normal bundle is a line bundle. It is not difficult to show that E intersects itself negatively. This means that its normal bundle possesses no holomorphic sections; E is the only smooth complex representative of its homology class in . (Suppose E could be perturbed off itself to another complex submanifold in the same class. Then the two submanifolds would intersect positively — as complex submanifolds always do — contradicting the negative self-intersection of E.) This is why the divisor is called exceptional.

Let V be some submanifold of X other than Z. If V is disjoint from Z, then it is essentially unaffected by blowing up along Z. However, if it intersects Z, then there are two distinct analogues of V in the blow-up . One is the proper (or strict) transform, which is the closure of ; its normal bundle in is typically different from that of V in X. The other is the total transform, which incorporates some or all of E; it is essentially the pullback of V in cohomology.