On the L^2-Hodge theory of Landau-Ginzburg models (I)

Abstract: Given a non-compact Calabi-Yau manifold X and a holomorphic function f on it with only compact critical locus, I will describe an L^2 theoretic approach to study the deformation theory of this Landau-Ginzburg model. Concretely, the notion of f-twisted Sobolev spaces is introduced when f satisfies an asymptotic condition and it is then used to prove the Hodge-to-de Rham degeneration property. This leads to a Frobenius manifold structure via the Barannikov-Kontsevich construction and unifies the Landau-Ginzburg and Calabi-Yau geometry.