  We make a proposal for calculating refined Gopakumar-Vafa numbers (GVN) on
elliptically fibered Calabi-Yau 3-folds based on refined holomorphic anomaly
equations. The key examples are smooth elliptic fibrations over (almost) Fano
surfaces. We include a detailed review of existing mathematical methods towards
defining and calculating the (unrefined) Gopakumar-Vafa invariants (GVI) and
the GVNs on compact Calabi-Yau 3-folds using moduli of stable sheaves, in a
language that should be accessible to physicists. In particular, we discuss the
dependence of the GVNs on the complex structure moduli and on the choice of an
orientation. We calculate the GVNs in many instances and compare the B-model
predictions with the geometric calculations. We also derive the modular anomaly
equations from the holomorphic anomaly equations by analyzing the quasi-modular
properties of the propagators. We speculate about the physical relevance of the
mathematical choices that can be made for the orientation.
