Stability and equivalence of admissible pairs of arbitrary dimension for a compactification of the moduli space of stable vector bundles

Moduli spaces of stable vector bundles and compactifications of these moduli spaces are closely related to Yang–Mills gauge field theory. This paper is devoted to finding an appropriate compactification of the moduli space of stable vector bundles on an algebraic variety of dimension $\ge 2$. We consider admissible pairs $((\widetilde S, \widetilde L), \widetilde E)$, each of which consists of an $N$-dimensional admissible scheme $\widetilde S$ of some class with a certain ample line bundle $\widetilde L$ and of a vector bundle $\widetilde E$. An admissible pair can be obtained by a transformation (called a resolution) of a torsion-free coherent sheaf $E$ on a nonsingular $N$-dimensional projective algebraic variety $S$ to a vector bundle $\widetilde E$ on a certain projective scheme $\widetilde S$. The notions of stability (semistability) for admissible pairs and of M-equivalence for admissible pairs in the multidimensional case are introduced. We also study relations of the stability (semistability) for admissible pairs to the classical stability (semistability) for coherent sheaves under the resolution and relations of the M-equivalence for semistable admissible pairs to the S-equivalence of coherent sheaves under the resolution. The obtained results are intended for constructing a compactification of the moduli space of stable vector bundles and an ambient moduli space of semistable admissible pairs.