Remarks on the Chern classes of Calabi-Yau moduli
  We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors. 
Kefeng Liu, Changyong Yin
math.DG