Counting special Lagrangian fibrations in twistor families of K3 surfaces
Abstract
The number of closed billiard trajectories in a rational-angled polygon grows quadratically in the length. This paper gives an analogue on K3 surfaces, by considering special Lagrangian tori. The analogue of the angle of a billiard trajectory is a point on a twistor sphere, and the number of directions admitting a special Lagrangian torus fibration with volume bounded by $ V $ grows like $ V^{20} $ with a power-saving term. Bergeron--Matheus have explicitly estimated the exponent of the error term as $ {20-\frac{4}{697633} }$. The counting result on K3 surfaces is deduced from a count of primitive isotropic vectors in indefinite lattices, which is in turn deduced from equidistribution results in homogeneous dynamics.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2016
- DOI:
- 10.48550/arXiv.1612.08684
- arXiv:
- arXiv:1612.08684
- Bibcode:
- 2016arXiv161208684F
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Algebraic Geometry;
- Mathematics - Differential Geometry;
- Mathematics - Dynamical Systems
- E-Print:
- 39 pages, 2 figures. Final version based on the referee report. To appear in Annales scientifiques de l'\'Ecole Normale Sup\'erieure. Appendix by Bergeron-Matheus at arXiv:1703.01746