A topological proof of Bloch's conjecture
The following is an outline of the structure of the thesis. • In the first chapter, after reviewing the Chern Weil construction of characteristic classes of bundles I introduce the constructions of secondary invariants given by Chern and Simons and by Cheeger and Simons . • In the second chapter I introduce Bloch's conjecture and give some of its motivations. In particular we define Deligne cohomology and its smooth analogue, and prove that the latter is isomorphic to Cheeger and Simons ring of differential characters. • In the third chapter I review Lefschetz Theorems and related results about the topology of smooth projective algebraic varieties, including an analysis of the local structure near a singular hyperplane section. • The fourth chapter is the heart the thesis. I prove the two main results Theorem 4.0.1 and Theorem 4.0.2, asserting that the rational three dimensional homology of a smooth projective algebraic variety is generated by the images of the fundamental classes of S2 x S1 and of connected sums of S1 x S2 under some appropriate maps. The rationality of the Chern-Cheeger-Simons class follows directly from this result. I also discuss briefly the difficulties which I encountered when trying to generalize the construction to give inductively generators for all the odd homology groups of smooth projective algebraic manifolds.