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The Hodge Conjecture is equivalent to a statement about conditions underwhich a complex vector bundle on a smooth complex projective variety admits aholomorphic structure. I advertise a class of abelian four-folds due to Mumfordwhere this approach could be tested. I construct explicit smooth vector bundles- which can in fact be constructed in terms of of smooth line bundles - whoseChern characters are given Hodge classes. An instanton connection on thesevector bundles would endow them with a holomorphic structure and thus provethat these classes are algebraic. I use complex multiplication to exhibitCayley cycles representing the given Hodge classes. I find alternate complexstructures with respect to which the given bundles are holomorphic, and closewith a suggestion (due to G. Tian) as to how this may possibly be put to use.

Spin(7) instantons and the Hodge Conjecture for certain abelian four-folds: a modest proposal