From real affine geometry to complex geometry

By Mark Gross, Bernd S Siebert. 2007

308 citations recorded by the source corpus.

We construct from a real ane 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.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/98977e3eb43f976085bc5add93b44ec7961f15b5

How researchers see this paper

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In Geometry and Topology

1. D. Auroux
2. L. Katzarkov
3. P. Aspinwall
4. G. Moore
5. T. Bridgeland
6. M. Abouzaid
7. Yukinobu Toda
8. Mark Gross

In Statistical and Nonlinear Physics

1. M. Abouzaid
2. L. Katzarkov
3. D. Auroux
4. G. Moore
5. P. Aspinwall
6. Balázs Szendrői
7. Mark Gross
8. Michael R Douglas

Citations in the corpus

• Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c} (2013, background): “uld give an enumerative meaning of mirror maps of Fermat hypersurfaces. It is motivated by the principle that (generalized) SYZ mirror should be automatically written in at coordinates. Gross-Siebert [GS11] made a conjecture of this type in the tropical world, namely the mirror they construct using ”
• Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “Unfortunately, there are many varieties that do not admit a well-understood embedding into a toric variety, such as the Deligne–Knudsen–Mumford moduli spaces Mg,n of stable n-marked curves for g > 0 (see [14] and [34]) or the toric degenerations arising in the GrossSiebert approach to mirror symmetr”
• Dirichlet Branes and Mirror Symmetry (2009, background): “(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].”