Extension across the low-rank locus as a special Lagrangian integral current

Determine whether each submanifold L_Q, consisting of balanced invertible matrix factorizations whose product has unitary polar factor Q, extends across the low-rank locus as a special Lagrangian integral current.

Background

For each unitary matrix Q, the paper defines L_Q as a special Lagrangian submanifold of the space of N-tuples of invertible complex matrices satisfying the balancedness equations. The authors then study its closure C_Q in the ambient space, where the common positive-definite factor may become positive semidefinite and the matrix components may lose rank.

The treatment of these closures is explicitly set-theoretic. The unresolved issue is whether the smooth special Lagrangian geometry of L_Q can be extended through the rank-deficient boundary in the framework of special Lagrangian integral currents.