Bounded Geometries on Hybrid Landau-Ginzburg models of Calabi-Yau complete intersections and $L^2$-Hodge Theory
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| Format: | Preprint |
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2025
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| _version_ | 1866911533102006272 |
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| author | Park, Jeehoon Yoo, Jaewon |
| author_facet | Park, Jeehoon Yoo, Jaewon |
| contents | 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 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 $H(V;\mathbb{C})$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_24218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded Geometries on Hybrid Landau-Ginzburg models of Calabi-Yau complete intersections and $L^2$-Hodge Theory Park, Jeehoon Yoo, Jaewon Algebraic Geometry 14J32 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 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 $H(V;\mathbb{C})$, 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. |
| title | Bounded Geometries on Hybrid Landau-Ginzburg models of Calabi-Yau complete intersections and $L^2$-Hodge Theory |
| topic | Algebraic Geometry 14J32 |
| url | https://arxiv.org/abs/2505.24218 |