  We previously developed an approach to Bialynicki-Birula theory for
holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and the concept
of virtual Morse-Bott indices for singular critical points of Hamiltonian
functions for the induced circle actions (see Feehan, arXiv:2206.14710). For
Hamiltonian functions of circle actions on closed, complex Kaehler manifolds,
the virtual Morse-Bott index coincides with the classical Morse-Bott index due
to Bott (1954) and Frankel (1959). A key principle in our approach is that
positivity of the virtual Morse-Bott index at a critical point of the
Hamiltonian function implies that the critical point cannot be a local minimum
even when that critical point is a singular point in the moduli space. In this
monograph, we consider our method in the context of the moduli space of
non-Abelian monopoles over a closed, complex, Kaehler surface. We use the
Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all
critical strata (Seiberg-Witten moduli subspaces) and we prove that these
indices are positive in a setting motivated by the conjecture that all closed,
smooth four-manifolds of Seiberg-Witten simple type obey the
Bogomolov-Miyaoka-Yau inequality.
