Nonexistence of first-type regulators for singular polarizations with non-Lagrangian isotropic fibers

A singular polarization ${\mathcal P}$ on a symplectic manifold $M$ assigns a complex subspace ${\mathcal P}_x\subset T_xM\otimes {\mathbb C}$ at each point $x\in M$. A fiber is isotropic when the symplectic form vanishes on it, and non-Lagrangian when it is not a Lagrangian subspace. A regulator of the first type is a function used to deform the polarization so that the deformed polarization becomes Kähler.

Nonexistence conjecture. There do not exist regulators of the first type for singular polarizations ${\mathcal P}$ such that there exist points $x\in M$ for which ${\mathcal P}_x$ is an isotropic non-Lagrangian subspace of $T_xM\otimes {\mathbb C}$.

This conjecture extends the preceding nonexistence theorem for real analytic polarizations arising from completely integrable systems on compact manifolds. Its general status is not specified in the source.