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Building on the results of Ref. \cite{faye2018phase}, which identified anantiferromagnetic and Kondo singlet phases on the Kondo-Hubbard square lattice,we use the variational cluster approximation (VCA) to investigate thecompetition between these phases on a two-dimensional triangular lattice with$120^{o}$ spin orientation. In addition to the antiferromagnetic exchangeinteraction $J_{\perp}$ between the localized (impurity) and conduction(itinerant) electrons, our model includes the local repulsion $U$ of theconduction electrons and the Heisenberg interaction $J_H$ between theimpurities. At half-filling, we obtain the quantum phase diagrams in bothplanes $(J_{\perp}, U J_{\perp})$ and $(J_{\perp}, J_{H})$. We identify along-range, three-sublattice, spiral magnetic order which dominates the phasediagrams for small $J_{\perp}$ and moderate $U$, while a Kondo singlet phasebecomes more stable at large $J_{\perp}$. The transition from the spiralmagnetic order to the Kondo singlet phase is a second-order phase transition.In the $(J_{\perp}, J_{H})$ plane, we observe that the effect of $J_H$ is toreduce the Kondo singlet phase, giving more room to the spiral magnetic orderphase. It also introduces some small magnetic oscillations of the spiralmagnetic order parameter. At finite doping and when spiral magnetism isignored, we find superconductivity with symmetry order parameter $d+id$, whichbreaks time reversal symmetry. The superconducting order parameter has a domecentered at around $5\%$ hole doping, and its amplitude decreases withincreasing $J_{\perp}$. We show that spiral magnetism can coexist with $d+id$state and that superconductivity is suppressed, indicating that these twophases are in competition.

Spiral magnetism and chiral superconductivity in Kondo-Hubbard triangular lattice model