  We introduce a regularized theta lift for reductive dual pairs of the form
$(Sp_4,O(V))$ with $V$ a quadratic vector space over a totally real number
field $F$. The lift takes values in the space of $(1,1)$-currents on the
Shimura variety attached to $GSpin(V)$, and we prove that its values are
cohomologous to currents given by integration on special divisors against
automorphic Green functions. In the second part to this paper, we will show how
to evaluate the regularized theta lift on differential forms obtained as usual
(non-regularized) theta lifts.
