Geodesic completeness of the affine connection on compact integral-integral affine bases

Determine whether the torsion-free flat affine connection induced by an integral-integral affine atlas on a compact manifold—specifically the compact base B of a Lagrangian torus fibration with integral-integral affine structure—is geodesically complete.

Background

In developing their Downstairs Theorem relating volume to lattice points on integral-integral affine manifolds, the authors consider the base B of a Lagrangian torus fibration, which acquires an integral-integral affine structure.

Despite B being compact, the authors point out that the geodesic completeness of the induced affine connection is unknown. They identify this as a special case of the Markus conjecture, which concerns completeness for closed affine manifolds whose transition maps have determinant ±1.