  The Hodge Conjecture is equivalent to a statement about conditions under
which a complex vector bundle on a smooth complex projective variety admits a
holomorphic structure. I advertise a class of abelian four-folds due to Mumford
where this approach could be tested. I construct explicit smooth vector bundles
- which can in fact be constructed in terms of of smooth line bundles - whose
Chern characters are given Hodge classes. An instanton connection on these
vector bundles would endow them with a holomorphic structure and thus prove
that these classes are algebraic. I use complex multiplication to exhibit
Cayley cycles representing the given Hodge classes. I find alternate complex
structures with respect to which the given bundles are holomorphic, and close
with a suggestion (due to G. Tian) as to how this may possibly be put to use.
