Primitive forms via polyvector fields

Understand

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.