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In this paper we describe the differential geometric properties of a quantumscalar field in terms of Hida-Malliavin calculus. We focus our attention on therole played by the different mathematical structures of the Cauchy hypersurfacewhich constitutes the domain of the classical fields and the geometricalstructures that the classical field phase-space is endowed with. We argue thatthe classical phase space must be modelled over the strong dual of a NuclearFrech\`et space and it is endowed with a complex structure, which allows todefine a K\"ahler structure on it using the canonical symplectic structure.This mathematical structure determines all required ingredients to describequantum objects through a geometric quantization procedure, analogous to theusual construction on a classical mechanics phase space. We will discuss indetail two different polarizations, one complex and one real; and will comparethree different quantizations frequently used in Physics: the usualSchr\"odinger quantization, the analogous one using the field-momenta insteadof the fields; and holomorphic quantization based on creation and annihilationoperators. We will see how the necessary ingredients of the quantizationprocedures take a simple form when written in terms of Malliavin tools. In thesubsequent paper, these tools will be used to characterize the set of quantumstates as a Poisson manifold and the unitary dynamics generated bySchr\"odinger formalism as integral curves of a Hamiltonian vector field.

Geometric flavours of Quantum Field theory on a Cauchy hypersurface. Part I: Geometric quantization