  The Dirac-Dolbeault operator for a compact K\"ahler manifold is a special
case of a Dirac operator. The Green function for the Dirac Laplacian over a
Riemannian manifold with boundary allows to express the values of the sections
of the Dirac bundle in terms of the values on the boundary, extending the mean
value theorem of harmonic analysis. Utilizing this representation and the
Nash-Moser generalized inverse function theorem we prove the existence of
complex submanifolds of a complex projective manifold satisfying globally a
certain partial differential equation under a certain injectivity assumption.
Next, we show the existence of complex submanifolds whose fundamental classes
span the rational Hodge classes, proving the Hodge conjecture for complex
projective manifolds.
