# 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.