  This is the first part in a series of papers on counting surfaces on
Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we
introduce \emph{two} types of moduli spaces of stable pairs. We show that all
three moduli spaces are related by GIT wall-crossing and parametrize stable
objects in the bounded derived category.
  We construct \emph{reduced} Oh-Thomas virtual cycles on the moduli spaces via
Kiem-Li cosection localization and prove that they are deformation invariant
along Hodge loci. As an application, we show that the variational Hodge
conjecture holds for any family of Calabi-Yau 4-folds supporting a non-zero
reduced virtual cycle.
