  We develop a complex differential geometric approach to the theory of higher
residues and primitive forms from the viewpoint of Kodaira-Spencer gauge
theory, unifying the semi-infinite period maps for Calabi-Yau models and
Landau-Ginzburg models. We give an explicit perturbative construction of
primitive forms with respect to opposite filtrations and primitive elements.
This leads to a concrete algorithm to compute the Taylor expansions of
primitive forms as well as the description of their moduli space for all
weighted homogenous cases. As an example, we present unknown perturbative
expressions for the primitive form of E_12 singularity and illustrate its
application to Landau-Ginzburg mirror symmetry with FJRW-theory.
