an:02210045
Zbl 1085.14035
Grassi, Michele
Self-dual manifolds and mirror symmetry for the quintic threefold
EN
Asian J. Math. 9, No. 1, 79-101 (2005).
00117976
2005
j
14J32 14M25
reflexive polytopes; Gromov-Hausdorff distance
The author describes a way to geometrically interpolate between the large K??hler structure limit of the K??hler moduli space of the anticanonical divisor in \(\mathbb{P}^n\) and a large complex structure limit of the complex structure moduli space of the mirror partner given by the \textit{B. R. Greene, M. R. Plesser} orbifold construction [Duality in Calabi-Yau moduli space, Nucl. Phys. B338, 15--37 (1990)]. This is achieved by constructing a two-dimensional family of smooth manifolds \(\mathbb{X}_{\rho_1,\rho_2}\) of real dimension \((3(n-1) + 2)\) endowed with a ``weakly self-dual'' (WSD) structure. A WSD structure consists of three closed 2-forms and a Riemannian metric satisfying certain integrability and compatibility conditions. Taking appropriate limiting values for \(\rho_1\) and \(\rho_2\), the manifolds \(\mathbb{X}_{\rho_1,\rho_2}\) approach the large K??hler structure limit of the anticanonical divisor and the large complex structure limit of the mirror in a normalized Gromov-Hausdorff sense.
The construction starts with the fiber product \((\mathbb{C}^*)^{n+1} \times_\mu (\mathbb{C}^*)^{n+1}\) over \(\mathbb{R}^{n+1}\), where \(\mu: (\mathbb{C}^*)^{n+1} \rightarrow \mathbb{R}^{n+1}\) is the usual \(T^{n+1}\)-moment map given by rotating the coordinates. The WSD manifolds arise from a sort of ``polysymplectic reduction'' of \((\mathbb{C}^*)^{n+1} \times_\mu (\mathbb{C}^*)^{n+1}\) by a group action arising from the reflexive polytope construction of \(\mathbb{P}^n\) and its toric dual.
Edward Lee (Los Angeles)