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.diffraction and x-ray measurements of the triangular antiferromagnet AgNiO 2 also display evidence of charge order in the Ni triangular lattice while the system remains metallic [59,60]. However, although much progress has been made in the understanding of TLMs many issues are still open. In particular, the extent to which charge or spin fluctuations [61] can act in expanding the range of pairing possibilities is much less understood.Spurred by these important advances, we investigate here ordering modulations pinned by impurities in TLMs as well as the competition between superconductivity and charge-density waves (CDWs) in + d idor fwave superconductors, with possible relevance to NCO and organic superconductors. Based on the − ′ − − t t J V model, we theoretically demonstrate that the appropriate breaking of the commensurability can induce charge fluctuations and show that charge-density modulations with a period of roughly 4.5 3 lattice constants around impurities near an incommensurate filling can appear in TLMs. To the best of our knowledge, these interesting modulations have not been reported previously. Moreover, we also consider the competition between CDW and superconductivity in + d idor f-wave superconductors and explore the local density of states around the impurity. It is found that low-energy quasiparticle resonance states appear at the impurity site in both pairing cases and meanwhile the asymmetric SC coherence peaks are strongly suppressed. The LDOS spectra gradually recover the bulk feature at sites far from the impurity, which show a V-shape in the f-wave case. In contrast, the far-site LDOS of the + d idstate shows a doping-dependent gap anisotropy, which is fully gapped at low doping levels but gradually evolves into close-to-nodal gap features as the doping increases. These results may be helpful in understanding some exotic properties of cobaltate and organic materials.The rest of the paper is organized as follows. In section 2, we formulate the physical model and introduce Bogoliubov-de Gennes equations. In section 3, we consider the charge-density modulations in the non-SC state. We then discuss in section 4 the competition between CDW and superconductivity in + d idor f-wave superconductors with an incommensurate or 1/3 filling. The results of the local ...