In this article, we investigate the global well-posedness of fractional-order nonlinear Schrödinger equations (FNLS) \( i\partial _t u + (-\Delta _g)^{\frac{\sigma }{2}}u = -|u|^2u, \quad \sigma \in (d-1,d] \) on an arbitrary d-dimensional boundaryless compact manifold M with initial data \(u_0 \in H^s(M)\) with \( s > \frac{\sigma ^2 + 2\sigma d - 3\sigma - d^2 + d}{6\sigma - 2d - 2} \in \left( \frac{d-\sigma }{2}, \frac{\sigma }{2} \right) . \) First, we employ the semi-classical analysis method together with refined bilinear oscillatory integral estimates to derive bilinear Strichartz estimates for the space-time slab \([0,1] \times M_\lambda \) , where \(M_\lambda \) is the re-scaled manifold of M. We then combine these bilinear estimates with multilinear eigenfunction estimates from [36] and the I-method from [14] to prove global well-posedness below the energy space \(H^{\frac{\sigma }{2}}(M)\) .
