Topologically Trivial Sheaves on Curves with Simplest Singularities

Topologically trivial vector bundles on nodal curves were considered in Tyurin's book Quantization, Classical and Quantum Field Theory, and Theta Functions. In the present paper, a compactification of the moduli of such vector bundles by topologically trivial torsion-free sheaves is constructed, and a stability criterion for topologically trivial sheaves of ranks $1$ and $2$ is given.