Wall crossing for the \(\Gamma \)-equivariant bundles on surfaces

In this article, we identify natural wall and chamber structures in the Kähler cone of a complex smooth projective surface Y. We study the variation of the Gieseker compactifications of the moduli spaces of rank 2 \(\chi \) $\chi$ -stable \(\Gamma \) $\mathrm{\Gamma }$ -equivariant sheaves on Y with respect to two different ample line bundles, H and \(H'\) $H^{\prime }$ in the adjacent chambers. We establish the existence of canonical rational morphisms among the Gieseker compactifications and provide explicit descriptions of these maps, which restrict to the identity on some Zariski open subset. Our results extend to the study of parabolic bundles, offering new insights into the structure and geometry of moduli spaces.