Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces.(English)Zbl 1539.14077
The principal motivation of this paper is the Strominger-Yau-Zaslow conjecture, which concerns finding a special Lagrangian torus fibration on the generic region of Calabi-Yau manifolds closed to the large complex structure limit. The main challenge is to understand the limiting behaviour of the Calabi-Yau metrics, and it is generally expected that the potentially theoretic limit of the Calabi-Yau metric is a solution to the real Monge-Ampère equation on the essential skeleton associated to the degeneration family. However, formulating this real Monge-Ampère equation is challenging, because the essential skeleton is a priori only a simplicial complex, where the real Monge-Ampère equation cannot be defined in the classical way.
The main contribution of this paper is to introduce a variational perspective for the Kontorovich functional, which is motivated from optimal transport theory. In a highly symmetric case, the authors show that this perspective provides the desirable properties of the real Monge-Ampère equation. They also provide a number of pathological examples when the symmetry is not satisfied. As applications of their main results, they recover the known results about the Strominger-Yau-Zaslow conjecture in the Fermat hypersurface case, and show that their solution reproduces the solution to the non-archimedean Monge-Ampère equation.
The main contribution of this paper is to introduce a variational perspective for the Kontorovich functional, which is motivated from optimal transport theory. In a highly symmetric case, the authors show that this perspective provides the desirable properties of the real Monge-Ampère equation. They also provide a number of pathological examples when the symmetry is not satisfied. As applications of their main results, they recover the known results about the Strominger-Yau-Zaslow conjecture in the Fermat hypersurface case, and show that their solution reproduces the solution to the non-archimedean Monge-Ampère equation.
Reviewer: Yang Li (Cambridge, MA)
MSC:
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 32P05 | Non-Archimedean analysis |
| 32Q25 | Calabi-Yau theory (complex-analytic aspects) |
| 53A15 | Affine differential geometry |
| 05E14 | Combinatorial aspects of algebraic geometry |
| 14T90 | Applications of tropical geometry |
Keywords:
Calabi-Yau manifolds;SYZ conjecture;essential skeleton;Monge-Ampère equations;special Lagrangian fibrationCite
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