On arborealization, Maslov data, and lack thereof

For a Weinstein manifold, we compare and contrast the properties of admitting an arboreal skeleton and admitting Maslov data. Both properties are implied by the existence of a polarization, for which a basic obstruction is that the odd Chern classes are 2-torsion. In a similar spirit we establish cohomological obstructions to the existence of arboreal skeleta and to the existence of Maslov data, exhibiting concrete examples to illustrate their failure. For instance, we show that complements of smooth anti-canonical divisors in complex projective space may fail to admit either arboreal skeleta or Maslov data. We also exhibit an example of a Weinstein manifold which admits Maslov data but does not admit an arboreal skeleton.