Bounded Geometries on Hybrid Landau-Ginzburg models of Calabi-Yau complete intersections and $L^2$-Hodge Theory
Jeehoon Park, Jaewon Yoo
TL;DR
The paper constructs a hybrid Landau–Ginzburg model for Calabi–Yau complete intersections via the Cayley trick and proves the existence of a complete Kähler metric yielding a bounded Calabi–Yau geometry with a strongly elliptic superpotential. This enables the Li–Wen $L^2$-Hodge framework to produce a Frobenius manifold on the twisted de Rham cohomology $H(PV(X_{CY}),\overline{\partial}_W)$, which is shown to be isomorphic to $H(V(\underline{W});\mathbb{C})$, thereby giving a new $L^2$–Hodge theoretic construction of a Frobenius structure on the CY complete intersection cohomology. The work further develops a spectral-sequence analysis (analytic and algebraic) to bridge the LG and CY sides, and provides a detailed geometric construction ensuring bounded geometry, strong ellipticity, and a simultaneous toric blow-up framework relating $X_{CY}$ and $X_{LG}$. These results broaden the toolkit for constructing Frobenius manifolds in noncompact Calabi–Yau settings and connect LG/CY cohomology to classical CY cohomology through $L^2$-Hodge theory. The approach combines differential-analytic methods with algebraic techniques (Adolphson–Sperber, GAGA) to obtain a robust cohomological correspondence with potential applications in mirror symmetry and deformation theory.
Abstract
Given a Calabi-Yau smooth projective complete intersection variety $V$ over $\C$, 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 Kähler metric $\mathfrak{g}$ and a bounded Calabi-Yau volume form $Ω$ on $X_{CY}$ such that $(X_{CY},\mathfrak{g}, Ω)$ 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 $(X_{CY},\mathfrak{g}, Ω)$ 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 $H(V;\C)$, which results in a new $L^2$-Hodge theoretic construction of a Frobenius manifold structure on $H(V;\C)$.