I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature.
Let $X$ be a Kähler surface. Let $\mathscr{M}(c_1, r, c_2)$ be the moduli space of all slope stable complex vector bundles $V$ over $X$ of rank $r$ with $c_1(V) = c_1$ and $c_2(V) = c_2$. Donaldson-Uhlenbeck-Yau show that there is an isomorphism $$\mathscr{M}(0,2,c_2) \leftrightarrow \{\text{ASD } SU(2) \text{ connections over } X\}/\text{ gauge group}.$$ Taubes also has shown that for an arbitrary $4-$manifold $M$ and $g$ is any Riemann metric on $M$ and $P$ is an $SU(2)-$principal bundle over $M$, the moduli space $\mathscr{M}(P, c_2,g)$ of ASD $SU(2)$ connections of $P$ is non-empty if $c_2 = c_2(P)$ is sufficiently large. Thus in the Kähler case, for sufficiently large $c_2$, $\mathscr{M}(0,2,c_2) \neq \emptyset$.
My questions:
- What exactly is this sufficiently large value of $c_2$? Is there a precise bound?
- Is there an analogous statement for higher rank case when $r>2$?