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

Counting surfaces on Calabi-Yau 4-folds I: Foundations