There is an introduction to the BCOV theory from math perspective in [this artcle][1] by Kanazawa and Zhou. In the case $h^{1,1}=1$, it is checked in page 16 that the $\mathcal{F}_1$ (non-holomorphic object) satisfies the anomaly equation. The general case should follow in a smiler manner. 


  [1]: http://arxiv.org/abs/1409.4105