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.

39 pages, 2 figures. Final version based on the referee report. To appear in Annales scientifiques de l'École Normale Supérieure. Appendix by Bergeron-Matheus at arXiv:1703.01746

References in corpus (2)

• Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow
• On special Lagrangian fibrations in generic twistor families of K3 surfaces

Cited by in corpus (1)

• Tropical Dynamics of Area-Preserving Maps