The nature of superconductivity in newly discovered kagome materials,
AV_33Sb_55
(A = K, Rb, Cs), has been a subject of intense debate. Recent
experiments suggest the presence of orbital current order on top of the
charge density wave (CDW) and superconductivity. Since the orbital
current order breaks time-reversal symmetry, it may fundamentally affect
possible superconducting states. In this work, we investigate the mutual
influence between the orbital current order and superconductivity in
kagome metal with characteristic van Hove singularity (vHS). By
explicitly deriving the Landau-Ginzburg theory, we classify possible
orbital current order and superconductivity. It turns out that distinct
unconventional superconductivities are expected, depending on the
orbital current ordering types. Thus, this information can be used to
infer the superconducting order parameter when the orbital current order
is identified and vice versa. We also discuss possible experiments that
may distinguish such superconducting states coexisting with the orbital
current order.