Toric degenerations and Batyrev-Borisov duality

By Mark Gross. 2004

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This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I" (math.AG/0309070). In this paper, I consider the construction as it applies to the Batyrev-Borisov construction for complete intersections in toric varieties. Given a pair of reflexive polytopes with nef decompositions dual to each other, and given polarizations on the corresponding toric varieties, we construct examples of toric degenerations and their dual intersection complexes, which are affine manifolds with singularities. We show these affine manifolds are related by the discrete Legendre transform, thus showing that the Batyrev-Borisov construction is a special case of our more general construction. The description of the dual intersection complexes in terms of the combinatoricsof the setup generalises work of Haase and Zharkov in the toric hypersurface case, and similar work of Ruan. In particular, this gives a topological description of the Strominger-Yau-Zaslow fibrations on complete intersections in toric varieties.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4a4d229823b61caecc90c015f290dd3e539f2c1f

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Citations in the corpus

An invitation to toric degenerations (2008, background): “le to think of is a degeneration of a quartic surface in P3 to the union of the coordinate hyperplanes. More generally, any Calabi-Yau complete intersection in a toric variety has toric degenerations [Gr2]. Thus the notion of toric degeneration is a very versatile one, conjecturally giving all defor”
Lagrangian 3-torus fibrations (2006, methodology): “The latter statement is the content of [8]Theorem 0.”
Mirror Symmetry via Logarithmic Degeneration Data I (2003, background): “In [ 9 ], the first author checked that this agrees with Batyrev-Borisov duality.”
Theta functions on varieties with effective anti-canonical class (2016, background): “ns is a natural one from the point of view of mirror symmetry, as the mirror of a toric degeneration is a toric degeneration. This point of view incorporates, for example, all Batyrev-Borisov mirrors [Gr1]. However, it is not clear how generally toric degenerations can be constructed given Yη. On th”
From real affine geometry to complex geometry (2007, background): “Furthermore, such integral affine manifolds arise naturally from boundaries ofreflexive polytopes [ Gr3 ][HaZh].”
Mirror Symmetry via Logarithmic Degeneration Data II (2007, background): “In [10], the first author checked that this agrees with Batyrev-Borisov duality.”