Topology of algebraic varieties

Abstract

Throughout this appendix we work with projective varieties X ⊂ ℙ ℂ N —that is, with complex projective varieties. We can also view such a variety as a complex analytic, or holomorphic, subvariety of ℙ ℂ N —that is, a subset locally defined by the vanishing of analytic equations—or, if X is smooth, as a complex submanifold of ℙ N C . The topology induced from the standard topology on ℙ ℂ N , referred to as the classical , or sometimes analytic, topology, is much finer than the Zariski topology with which we have dealt in this text. Using it, we can consider geometric invariants of X such as the singular homology and cohomology groups H * ( X , ℤ) and H * ( X ,ℤ). In this appendix, we explain a little of what is known about such invariants. Throughout, when we speak of topological properties of X , we refer to the classical, or analytic, topology. GAGA theorems One might think that there would be many more holomorphic subvarieties of ℙ N than algebraic subvarieties, or that in passing from a smooth projective variety X over ℂ to its underlying complex manifold we would be losing information, since regular functions are holomorphic but not conversely. But this is not the case: Theorem C.1 (Chow). Every holomorphic subvariety of ℙ ℂ N is algebraic . See for example Griffiths and Harris [1994, Section I.3] for a proof. Many further results in this direction were proven in Serre [1955/1956]. These are collectively known as the GAGA theorems, after the name of Serre's paper (“Geometrie algebrique et geometrie analytique”). It follows immediately from Chow's theorem that if X and Y are projective varieties over C then any holomorphic map f : X → Y is algebraic (Proof: Apply Theorem C.1 to the graph Γ f ⊂ X × Y ). Not quite so immediate are the facts that any holomorphic vector bundle on a projective variety is algebraic and that if e is any such vector bundle on X then any global holomorphic section of e is algebraic. More generally, the Cech cohomology groups of e will be the same, whether computed for the sheaf of holomorphic sections of e in the analytic topology or the sheaf of regular sections in the Zariski topology.