Complex Manifolds

Abstract

In the preceding chapter we studied the topological space X(ℂ) associated with an arbitrary algebraic variety defined over the complex numbers ℂ. The example of nonsingular projective curves already gives a feeling for the extent to which X(ℂ) characterises the variety X. We proved that in this case the genus g of X is the unique invariant of the topological space X(ℂ). Thus we can say that the genus is the unique topological invariant of a nonsingular projective curve. The genus is undoubtedly an extremely important invariant of an algebraic curve, but it is very far from determining it. We saw in Chap. III, 6.6 that there are very many nonisomorphic curves of the same genus. The relation between a variety X and the topological space -X(ℂ) is similar in nature for higher dimensional varieties.