Lagrange torus fibrations of Calabi-Yau hypersurfaces in toric varieties and SYZ mirror symmetry conjecture. (English) Zbl 1071.14040
D’Hoker, Eric (ed.) et al., Mirror symmetry IV. Proceedings of the conference on strings, duality, and geometry, CRM, Montréal, Canada, 2000. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: International Press (ISBN 0-8218-3335-9/hbk). AMS/IP Stud. Adv. Math. 33, 33-55 (2002).
The paper under review is motivated by a geometric construction of mirror manifolds via special Lagrangian torus fibration. This construction was suggested by A. Strominger, S. Yau and E. Zaslow (SYZ) [Nucl. Phys. B 479, 243–259 (1996; Zbl 0896.14024)], and today it is commonly known as the SYZ construction (or the SYZ mirror conjecture). It proposes that a Calabi-Yau threefold \(X\) should admit a special Lagrangian torus fibration and the mirror manifold of \(X\) is the moduli space of special Lagrangian \(3\)-torus in \(X\) with a flat \(\text{U}(1)\) connection. In this paper, the author constructs Lagrangian torus fibrations of generic Calabi-Yau hypersurfaces in toric variety corresponding to a reflexive polytope. In doing so, the symplectic topological SYZ conjecture for Calabi-Yau hypersurfaces in toric variety is established.
Theorem. For a generic Calabi-Yau hypersurface \(X\) and its mirror Calabi-Yau hypersurface \(Y\) near their corresponding large complex limit and large radius limit, there exist corresponding Lagrangian torus fibrations \[ \begin{matrix} X_{\phi(b)}& \hookrightarrow & X\qquad\qquad &Y_{b} \hookrightarrow &Y\\ & &\downarrow\qquad\qquad & & \downarrow\\ & &\partial\Delta_v\qquad\qquad& & \partial\Delta_w\end{matrix} \] with singular locus \(\Gamma\subset \partial \Delta_v\) and \(\Gamma^{\prime}\subset\partial \Delta_w\), where \(\phi:\partial\Delta_w\to\partial\Delta_v\) is a natural homeomorphism, \(\phi(\Gamma^{\prime})=\Gamma\). For \(b\in\partial\Delta_w\setminus\Gamma^{\prime}\), the corresponding fibers \(X_{\phi(b)}\) and \(Y_b\) are naturally dual to each other.
The original SYZ mirror conjecture was stated without mention of singular locus, singular fibers and duality of singular fibers. Thus, the first step taken in this paper for establishing the SYZ mirror conjecture is to formulate it in a more precise form. Detailed construction of special Lagrangian torus fibrations are carried out for quintic Calabi-Yau threefold and its mirror partner.
For the entire collection see [Zbl 1011.00027].
Theorem. For a generic Calabi-Yau hypersurface \(X\) and its mirror Calabi-Yau hypersurface \(Y\) near their corresponding large complex limit and large radius limit, there exist corresponding Lagrangian torus fibrations \[ \begin{matrix} X_{\phi(b)}& \hookrightarrow & X\qquad\qquad &Y_{b} \hookrightarrow &Y\\ & &\downarrow\qquad\qquad & & \downarrow\\ & &\partial\Delta_v\qquad\qquad& & \partial\Delta_w\end{matrix} \] with singular locus \(\Gamma\subset \partial \Delta_v\) and \(\Gamma^{\prime}\subset\partial \Delta_w\), where \(\phi:\partial\Delta_w\to\partial\Delta_v\) is a natural homeomorphism, \(\phi(\Gamma^{\prime})=\Gamma\). For \(b\in\partial\Delta_w\setminus\Gamma^{\prime}\), the corresponding fibers \(X_{\phi(b)}\) and \(Y_b\) are naturally dual to each other.
The original SYZ mirror conjecture was stated without mention of singular locus, singular fibers and duality of singular fibers. Thus, the first step taken in this paper for establishing the SYZ mirror conjecture is to formulate it in a more precise form. Detailed construction of special Lagrangian torus fibrations are carried out for quintic Calabi-Yau threefold and its mirror partner.
For the entire collection see [Zbl 1011.00027].
Reviewer: Noriko Yui (Kingston)
MSC:
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 32Q25 | Calabi-Yau theory (complex-analytic aspects) |
| 53D12 | Lagrangian submanifolds; Maslov index |
