Bounded Geometries on Hybrid Landau-Ginzburg models of Calabi-Yau complete intersections and $L^2$-Hodge Theory

Abstract

Given a Calabi-Yau smooth projective complete intersection variety $V$ over $\mathbb{C}$, a hybrid Landau-Ginzburg (LG) model may be associated using the Cayley trick. This hybrid LG model comprises a non-compact Calabi-Yau manifold $X_{CY}$, and a holomorphic function $W$, defined on $X_{CY}$, such that the critical locus of $W$ is isomorphic to $V$. We construct a complete K\"ahler metric $\mathfrak{g}$ and a bounded Calabi-Yau volume form ${\Omega}$ on $X_{CY}$ such that $(X_{CY},\mathfrak{g}, {\Omega})$ is a bounded Calabi-Yau geometry (in fact, $(X_{CY},\mathfrak{g})$ is an asymptotically conical manifold) and the function $W$ is strongly elliptic; this enables us to apply the $L^2$-Hodge theory of Li-Wen \cite{LW} to $(X_{CY},\mathfrak{g}, {\Omega})$ and $W$, which leads to a Frobenius manifold structure on the twisted de Rham cohomology associated to $(X_{CY},W)$. Furthermore, we prove that this twisted de Rham cohomology is isomorphic to the de Rham cohomology $H(V;\mathbb{C})$, which results in a new $L^2$-Hodge theoretic construction of a Frobenius manifold structure on $H(V;\mathbb{C})$. This paper provides the first explicit geometric verification of Li-Wen's theory for genuine non-isolated, compact critical loci using hybrid Landau-Ginzburg models.