Counting special Lagrangian fibrations in twistor families of K3 surfaces

TL;DR

This paper establishes a quadratic growth rate for counting special Lagrangian tori in K3 surfaces, paralleling billiard trajectory counts, using lattice and homogeneous dynamics techniques.

Contribution

It introduces a novel counting method for special Lagrangian fibrations on K3 surfaces, connecting geometric, lattice, and dynamical systems approaches.

Findings

Growth rate of special Lagrangian tori is proportional to V^{20}.

Error term exponent explicitly estimated as 20 - 4/697633.

Counting reduces to primitive isotropic vectors in indefinite lattices.

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.