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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
##### Keywords:
reflexive polytopes; Gromov-Hausdorff distance
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