Virtual-class compatibility under the Kobayashi–Hitchin isomorphism

Let $(V,g)$ be a surface with a Kähler metric $g$, let $m\in H^2(V,\mathbb{Z})$, and let $\beta$ be a real closed $2$-form of type $(1,1)$. Let $\tau$ be a $Spin^c(4)$-structure representing the class $\mathfrak{c}_m$, and let $\mathcal{W}^{\tau}_{\beta}$ be the moduli space of solutions to the $\beta$-twisted Seiberg–Witten equations. Choose the canonical orientation data. Suppose that $(2m-k-[\beta])\cdot[\omega_g]<0$. Kobayashi–Hitchin virtual-class conjecture. The isomorphism

$$\kappa^+_m:\mathcal{W}^{\tau}_{\beta}\stackrel{\cong}{\longrightarrow}\operatorname{Hilb}^m_V$$

identifies $[\mathcal{W}^{\tau}_{\beta}]_{vir}$ with the image of the virtual class $[[\operatorname{Hilb}^m_V]]$ in $H_*(\operatorname{Hilb}^m_V,\mathbb{Z})$. This is the conceptual algebro-geometric statement intended to explain the proposed equality between Poincaré and Seiberg–Witten invariants; the source does not establish it in general.