  A topological defect in the form of the Abrikosov-Nielsen-Olesen vortex is
considered as a gauge-flux-carrying tube that is impenetrable for quantum
matter. The relativistic spinor matter field is quantized in the vortex
background in $2+1$-dimensional conical space-time which is a section
orthogonal to the vortex axis; the most general set of boundary conditions
ensuring the impenetrability of the vortex core is employed. We find the
induced vacuum current circulating around the vortex and the induced vacuum
magnetic field strength pointing along the vortex axis. The requirement of
finiteness and physical plausibility for the total induced vacuum magnetic flux
allows us to restrict the variety of admissible boundary conditions. The
dependence of the results on the transverse size of the vortex, as well as on
the vortex flux and the parameter of conicity, is elucidated.
