On the Topology of Complex Algebraic Maps

In this largely expository note we give some homological properties of algebraic maps of complex algebraic varieties which are rather surprising from the topological point of view. These include a generalisation to higher dimension of the invariant cycle theorem for maps to curves. These properties are all corollaries of a recent deep theorem of Deligne, Gabber, Beilinson, and Bernstein which is stated in §2. This theorem involves intersection homology and the derived category. One of our objects here is to popularize it by giving corollaries involving only ordinary homology. For this reason some readers may wish to begin with §3. §]. Intersection homology. For any complex algebraic variety V , let Db(v) be the algebraically c constructible bounded derived category of the category sheaves of Q-module on V. (Objects of Db(v) are bounded complexes of sheaves of q-modules on V that c are cohomologically locally constant on the strata for some stratification of V by complex algebraic submanifolds; see ([GM2], §l.]l). If ~'C D~(V) and U c V , Hk(u,~ ")(resp. H~(U,~')) denotes the hyper-cohomology (resp. hypercohomology with compact supports) of the restriction of S" = to U. If p E V , let T ° be the "open disk" of points at distance less than P E from p , where distance is the usual Euclidean distance using some local analytic embedding of a neighborhood of P in ~N. For =S'C Db(V)c , Hk(~,S ")= and H~(~,~') are independent of the choices for small enough c • A local system on a space U is a locally constant sheaf of Q-module on U .