Herb Clemens

Making the Hodge problem "more geometric"

Abstract: We replace the intermediate Jacobian variety J(X) of a hypersurface sections X of a complex projective manifold of W dimension 2n with a 'slightly larger' abelian algebraic group K(X). This construction allows us to create normal functions (Abel-Jacobi maps) which associate to an arbitrary element h of primitive (2n-2)-homology of hyperplane sections of X a point in K(X). This normal function takes its value in the subgroup J(X) if and only if h lies in the Noether-Lefschetz locus of algebraic homology classes.