Affine manifolds, log structures, and mirror symmetry

By Mark Gross, Bernd S Siebert. 2002

98 citations recorded by the source corpus.

This is an outline of work in progress concerning an algebro-geometric form of the Strominger-Yau-Zaslow conjecture. We introduce a limited type of degeneration of Calabi-Yau manifolds, which we call toric degenerations. For these, the degenerate fibre is a union of toric varieties meeting along toric strata. In addition, the total space must look locally toric outside of some singular set. This is a generalisation of both Mumford-type degenerations of abelian varieties and degenerations of Calabi-Yau hypersurfacesin toric varieties. From such a degeneration, we construct a dual intersection graph along with a singular affine structure. We also explain how to reverse this procedure, using logarithmic structures. We then use a discrete Legendre transform on these affine manifolds to recover mirror symmetry. Finally, we discuss the relationship between this construction and more classical forms of SYZ. The details of the ideas presented here will appear elsewhere.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f7bae464d8c00c7c5223ea18287d1a013bd5f6e1

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Citations in the corpus

Mirror Symmetry via Logarithmic Degeneration Data II (2007, background): “This has already been discussed in many places, see for example [27],[11],[26],[7].”
Toric degenerations and Batyrev-Borisov duality (2004, methodology): “(This manifold carries a complex structure also, see [3] for details.”
Dirichlet Branes and Mirror Symmetry (2009, methodology): “(Lacking of course is a fiber over 0, and ideally one would like to have a degenerate Calabi-Yau over 0, but that is a story for another day: see [205, 206, 207, 208].”
Quantum periods for 3-dimensional Fano manifolds (2013, methodology): “Such an approach is likely to construct Fano manifolds via deformation methods, as in the Gross–Siebert mirror symmetry program [25–27], as opposed to explicit descriptions in the style of Theorem A.”
From real affine geometry to complex geometry (2007, result): “The results of this paper, combined with the results of [GrSi4] as described in [ GrSi1 ], can be viewed as giving as complete a solution to this problem as one could hope for.”
Symplectic theory of completely integrable Hamiltonian systems (2011, unknown): “Interactions have begun to develop in the context of mirror symmetry, where the study of singular Lagrangian fibrations is relevant [54, 55, 53, 52, 15, 16, 17].”
Almost toric symplectic four-manifolds (2003, unknown): “As a byproduct we provide a geometric explanation for why a generic Lagrangian fibration over the two-sphere must have 24 singular fibers.”
Toric degenerations of toric varieties and tropical curves (2004, methodology): “In connection with the mirror symmetry program with M. Gross [ GrSi1 ], [GrSi2], the second author has long suspected that this counting problem should have a purely combinatorial counterpart in integral affine geometry (cf.”
Constructing integrable systems of semitoric type (2009, background): “These structures have been studied in the context of integrable systems (in particular by Zung [23]), but also became a central concept in the works by Symington [18], Symington-Leung [15] in the c ontext of symplectic geometry and topology, and by Gross-Siebert [13], [11], [12] and [10], among ot h”
Mirror Symmetry via Logarithmic Degeneration Data I (2003, background): “This has already been discussed in many places, see for example [25],[ 10 ],[24],[6].”