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Self-dual manifolds and mirror symmetry for the quintic threefold. (English) Zbl 1085.14035
The author describes a way to geometrically interpolate between the large Kähler structure limit of the Kähler 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 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 Kähler 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.

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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