Steady axisymmetric outflows originating at the hot coronal magnetosphere of a Schwarzschild black hole and surrounding accretion disk are studied in the framework of general relativistic magnetohydrodynamics (GRMHD). The assumption of meridional self-similarity is adopted for the construction of semi-analytical solutions of the GRMHD equations describing outflows close to the polar axis. In addition, it is assumed that relativistic effects related to the rotation of the black hole and the

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... a are negligible compared to the gravitational and other energetic terms. The constructed model allows us to extend previous MHD studies for coronal winds from young stars to spine jets from Active Galactic Nuclei surrounded by disk-driven outflows. The outflows are thermally driven and magnetically or thermally collimated. The collimation depends critically on an energetic integral measuring the efficiency of the magnetic rotator, similarly to the non relativistic case. It is also shown that relativistic effects affect quantitatively the depth of the gravitational well and the coronal temperature distribution in the launching region of the outflow. Similarly to previous analytical and numerical studies, relativistic effects tend to increase the efficiency of the thermal driving but reduce the effect of magnetic self-collimation.