ModMax black hole surrounded by perfect-fluid dark matter in Lorentz-violating Kalb-Ramond gravity

We investigate a ModMax black hole surrounded by perfect-fluid dark matter within the framework of Lorentz-violating Kalb-Ramond gravity. The model combines three physically distinct contributions: nonlinear electrodynamic corrections from the ModMax sector, Lorentz-symmetry-breaking effects induced by the background Kalb-Ramond field, and environmental modifications associated with the surrounding dark matter fluid. We obtain the corresponding static and spherically symmetric black hole geometry and analyze how the charge, ModMax parameter, Kalb-Ramond coupling, and dark matter parameter affect the horizon structure and thermodynamic behavior. In particular, we study the Hawking temperature, entropy, heat capacity, and Helmholtz free energy, showing that the combined effects of nonlinear electrodynamics and Lorentz violation may shift the extremal configuration, modify the thermal stability regions, and generate nontrivial phase behavior. The perfect fluid dark matter contribution introduces an additional logarithmic correction to the geometry, becoming especially relevant at intermediate radial scales. Our results indicate that ModMax electrodynamics can effectively screen the electric sector, while the Kalb-Ramond parameter amplifies the geometric deformation and changes the thermodynamic response of the system. These features suggest that black holes in Lorentz-violating backgrounds surrounded by dark matter provide a useful arena for probing deviations from standard charged black-hole thermodynamics.