  In this paper, we study the analytic properties of solutions to the
Vafa-Witten equation over a compact Kaehler manifold. Simple obstructions to
the existence of nontrivial solutions are identified. The gauge theoretical
compactness for the $\mathbb{C}^*$ invariant locus of the moduli space is shown
to behave similarly as the Hermitian-Yang-Mills connections. More generally,
this holds for solutions with uniformly bounded spectral covers such as
nilpotent solutions. When spectral covers are unbounded, we manage to take
limits of the renormalized Higgs fields which are intrinsically characterized
by the convergence of the associated spectral covers. This gives a simpler
proof for Taubes' results on rank two solutions over Kaehler surfaces together
with a new complex geometric interpretation. The moduli space of $SU(2)$
monopoles and some related examples are also discussed in the final section.
