2002/11/05 by Mark Gross, Gross, Mark, Bernd Siebert +1 · 2 citations
Mathematics · #14J32 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG #msc:14J32
paper · pdf · doi:10.48550/arxiv.math/0211094
27 pages, 8 figures. Submitted to Gokova 2002 Conference proceedings
openalex publication_date 2002/11/05 · arxiv created 2003/03/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.