  Chiral three-spin interactions can suppress long-range magnetic order and
stabilize quantum spin liquid states in frustrated lattices. We study a
spin-1/2 model on the kagome lattice involving a staggered three-spin
interaction $J_\chi$ in addition to Heisenberg exchange couplings $J_1$ on
nearest-neighbor bonds and $J_d$ across the diagonals of the hexagons. We
explore the phase diagram using a combination of a classical approach, parton
mean-field theory, and variational Monte Carlo methods. We obtain a variety of
noncoplanar magnetic orders, including a phase that interpolates between
cuboc-1 and cuboc-2 states. In the regime of dominant $J_{\chi}$, we find a
classically disordered region and argue that it may harbor a gapless chiral
spin liquid with a spinon Fermi surface. Our results show that the competition
between the staggered three-spin interaction and Heisenberg exchange
interactions gives rise to unusual ground states of spin systems.
