[en] We identify a novel type of periodic charge-density modulation in triangular-lattice materials near an incommensurate filling. These charge modulations are suppressed by a strong pairing interaction, leaving the pairing modulations with the same periodicity. We also explore the competition between charge-density waves and superconductivity in chiral $d+id$ or f-wave superconductors, and we discuss the implications for cobaltate and organic superconductors. Furthermore, we self-consistently calculate the local density of states around impurities and show that the gap anisotropy of the chiral $d+id$ state is doping-dependent, which naturally explains the close-to-nodal gap features of cobaltate superconductors as indicated by experiments. (paper)