**Background:** In his two papers from late 80s Kronheimer proved that any 4-dimensional ALE space is given by a hyperkahler quotient, say $X_{{\zeta_\mathbb{R}},{\zeta_\mathbb{C}}}(Q)$ where Q is a Dynkin graph of type ADE, and ${\zeta_\mathbb{R}},{\zeta_\mathbb{C}}$ are parameters of hyperkahler moment map that satisfy a certain genericity conditions. He also proves that, fixing a graph Q all these spaces are diffeomorphic to a space, call it $X(Q),$ which is given as the minimal resolution of $$\pi: X(Q)\rightarrow \mathbb{C}^2/G,$$ (where G is a certain subroup of $SU(2)$). From earlier work of Du Val we know the topology of $X_Q$ - its deformational rectract is the exceptional divisor $$\pi^{-1}(0)=\cup_{i\in Q^0} \mathbb{C}P^1_i$$ which is a Dynkin Q tree of spheres that intersect transversely according to the graph, and whose self-intersections are -2. In particular, $$H_2(X(Q))=\oplus_{i\in Q^0} [\mathbb{C}P^1_i].$$

**Now the question:** It seemed to me that Kronheimer also proves the following: In the ALE space $X_{{\xi_\mathbb{R}},0}(Q)$ the $\omega_I$-volumes of those exeptional spheres are given
\begin{equation} \tag{1} \langle \omega_I, [\mathbb{C}P^1_i] \rangle= \zeta_\mathbb{R}^i
\end{equation}
exactly by the components of the real moment map parameters $\zeta_\mathbb{R}=(\zeta_{\mathbb{R}}^i)_{i\in{Q^0}}.$ Now this seems to be false, as those exceptional spheres are $\omega_I$-symplectic, hence their $\omega_I$-volumes are positive, whereas moment parameters $\zeta_{\mathbb{R}}^i$ can be negative.

So, does anyone knows ''the cure'' to the formula (1) to make it true?