From real affine geometry to complex geometry

Abstract

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.

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      VOLUME = {479},
      YEAR = {1996},
      NUMBER = {1-2},
      PAGES = {243--259},
      ISSN = {0550-3213},
      CODEN = {NUPBBO},
      MRCLASS = {32J17 (14J32 32J81 81T30)},
      MRNUMBER = {1429831},
      MRREVIEWER = {Mark Gross},
      DOI = {10.1016/0550-3213(96)00434-8},
      ZBLNUMBER = {0896.14024},
      }

Authors

Mark Gross

Department of Mathematics
University of California, San Diego
La Jolla, CA 92093-0112

Bernd Siebert

Department Mathematik
Universität Hamburg
20146 Hamburg
Germany