The topology of complex projective varieties after S. Lefschetz

Abstract

AFTER THE topology of complex algebraic curves, i.e. the genus of Riemannian surfaces, had been understood mathematicians like Picard [12] and PoincarC [ 12a] went on to the next dimension and began to investigate the topology of complex algebraic surfaces. From 1915 on Lefschetz continued their work and extended it to higher dimensional varieties. In 1924 he published his famous exposition [L] of this work. When it was written knowledge of topology was still primitive and Lefschetz “made use most uncritically of early topology g la PoincarC and even of his own later developments”?. This makes it nowadays rather difficult to understand the topological parts of [L] properly. But that is not the only difficulty: Implicitly Lefschetz quite often appeals to geometric intuition where we would like to see a more precise argument. Thus there is some temptation to discard Lefschetz’s original “proofs” and adopt instead the more recent methods which have been employed to obtain many of his results, using Hodge’s theory of harmonic differential forms or Morse theory or sheaf theory and spectral sequences. But none of these very elegant methods yields Lefschetz’s full geometric insight, e.g. they do not show us the famous “vanishing cycles”. The first attempt to rewrite the topological part of [L] using modern singular homology theory was made more than twenty years ago by Wallace [16]. But the details of his presentation are too complicated to popularize Lefschetz’s original methods. Wallace leaves the realm of algebraic geometry far too early when he makes Lefschetz’s intuitive arguments precise. Furthermore he does not give a complete picture of Lefschetz’s achievements. In the following I make a new attempt to present Lefschetz’s almost sixty year old investigations rigorously but as geometrically as he did in [L]. For topologists Lefschetz is usually interesting for the work he did in pure topology after he had completed [L]. But [Ll has at least “a unique historical interest in being almost the first account of the topology of a construct of importance in general mathematics which is not trivial” (Hedge). We may furthermore speculate how much of the contributions of Poincare, Lefschetz and others to algebraic topology we owe to the difficulties they encountered with the topology of algebraic varieties. The necessary prerequisites in algebraic geometry can be found in the first two chapters of Shafarevich’s book[l3]. The main tool from differential topology is Ehresmann’s fibration theorem, which for the convenience of the reader is stated in 3.0. (Strangely enough this theorem is not included in the standard textbooks.) As far as homology theory is concerned a textbook like Dold’s[6] will amply suffice. Furthermore some basic facts about the fundamental group and the homotopy lifting theorem for fibre bundles will be used.