$L^2$-Hodge theoretic construction of Frobenius manifolds for Calabi-Yau smooth projective hypersurfaces
Jeehoon Park, Jaewon Yoo
TL;DR
The article develops an $\!L^2$-Hodge theoretic construction of a Frobenius manifold on the cohomology of a Calabi–Yau hypersurface by applying Li–Wen's framework to a crepant resolution $X_{\mathrm{CY}}$ of $\mathbb{C}^n/\mu_n$ with potential $W$, and it proves an isomorphism $H(\mathrm{PV}(X_{\mathrm{CY}}), \overline{\partial}_W) \cong H(V; \mathbb{C})$. Using two spectral sequences, the authors compute $H^q(\mathrm{PV}(X_{\mathrm{CY}}), \overline{\partial}_W)$ and show it matches the cohomology of the Calabi–Yau hypersurface $V$, enabling an $L^2$-Hodge Frobenius structure on $H(V)$. They then compare this LW Frobenius algebra with the Barannikov–Kontsevich (BK) Frobenius algebra on $H(V)$, constructing a ring isomorphism that identifies the pairings up to a nonzero scalar. The results illuminate how two dGBV-based approaches to Frobenius structures on $H(V)$—one via a Landau–Ginzburg model with compact critical locus and another via Barannikov–Kontsevich’s formal theory—are closely related though not canonically identical, with implications for mirror symmetry and the geometry of CY hypersurfaces. Overall, the work provides explicit cohomology calculations, a precise algebraic comparison, and a discussion of when the corresponding Frobenius manifolds may or may not coincide.
Abstract
We provide a new $L^2$-Hodge theoretic construction of a Frobenius manifold structure on the cohomology of a Calabi-Yau smooth projective hypersurface $V$, using Li-Wen's $L^2$-Hodge theory [9] of a Landau-Ginzburg model with compact critical locus $V$. We also give a precise comparison result between the current construction and Barannikov-Kontsevich's construction [2] of the Frobenius manifold structure on the cohomology of $V$.