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It is known that angular momentum (AM) is an important characteristic of light, which can be separated into the spin (SAM) and orbital parts (OAM). The dynamical properties of the spin and orbital angular momentums are determined by the polarization and spatial degrees of freedom of light. In addition to optical vortex beams possessing spatial polarization and phase singularities, optical fibers can be used to generate and propagate optical modes with the orbital and spin parts of the angular momentum. In this paper, using the example of hollow-core fibers, we demonstrate the fact that their leaky air core modes also have an orbital part of AM in the case of circular polarization arising from the spin–orbit interaction of the air core modes. The reason for the appearance of AM is the leakage of the air core mode energy.