From real affine geometry to complex geometry

• 成果类型: Article
• 署名作者: Gross, Mark; Siebert, Bernd
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.3.1
• 发表日期: 2011
• 页码: 1301-1428
• 关键词: logarithmic degeneration data; mirror symmetry; toric degenerations; VARIETIES; CONSTRUCTION

摘要

We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect that our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods.