Mirror Symmetry via Logarithmic Degeneration Data I

By Mark Gross, Bernd S Siebert. 2003

256 citations recorded by the source corpus.

This paper is the first arising from our project announced in math.AG/0211094,"Affine manifolds, log structures, and mirror symmetry."We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certain sorts of degenerations of Calabi-Yau manifolds a log Calabi-Yau space, which is a log structure on the degenerate fibre. Then many statements about mirror symmetry which one hopes to be true for the general fibre should first be proved for this log CY space. In this paper we begin by discussing affine manifolds with singularities. Given such an affine manifold along with a polyhedral decomposition, we show how to construct a scheme consisting of a union of toric varieties. In certain non-degenerate cases, we can also construct log structures on these schemes. Conversely, given certain sorts of degenerations, one can build an affine manifold with singularities structure on the dual intersection complex of the degeneration. Mirror symmetry is then obtained as a discrete Legendre transform on these affine manifolds, thus providing an algebro-geometrization of the Strominger-Yau-Zaslow conjecture. The deepest result of this paper shows an isomorphism between log complex moduli of a log CY space and log Kahler moduli of its mirror.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8173e3d8c667f16c210fdec4405e900a4cb9404a

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Citations in the corpus

Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces (2012, methodology): “In general, intersections between walls lead, via a “scattering” phenomenon, to an infinite number of higher-order instanton corrections; it is conjectured that these Floer-theoretic corrections can be determined using the machinery developed by Kontsevich-Soibelman [31, 32] and Gross-Siebert [21, 2”
Towards Mirror Symmetry for Varieties of General Type (2012, background): “e with discriminant f= 0 which is also the singular locus of the total space of the family. This local Calabi-Yau is an essential building block of the toric degenerations studied by Gross-Siebert in [GS03]. In [AAK], the authors work out an understanding of mirror symmetry for varieties of general ”
Family Floer cohomology and mirror symmetry (2014, background): “The importance of the sheaf Aff was noted by Gross and Siebert [18], who related it to classical invariants of affine structures.”
Toric degenerations and Batyrev-Borisov duality (2004, methodology): “As we try to be as self-contained as possible, so that the reader need only consult [4] from time to time, in §1 we review the notion of toric degeneration and dual intersection complex.”
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].”
Lagrangian 3-torus fibrations (2006, methodology): “Our notion of simplicity can also be generalized to higher dimensions, though for n > 3 it may no longer coincide with the notion of simplicity in the sense of Gross and Siebert [14].”
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”
From real affine geometry to complex geometry (2007, methodology): “It is shown in [ GrSi2 ], §1.5 that this transformation has the following form:”
Collapsing of abelian fibred Calabi-Yau manifolds (2011, background): “This became known as the SYZ conjecture, and has generated an immense amount of work, see for example [16, 17, 18, 25] and references therein.”