# Kronheimer's results on ALE spaces as hyperkahler quotients

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 $$$$\tag{1} \langle \omega_I, [\mathbb{C}P^1_i] \rangle= \zeta_\mathbb{R}^i$$$$ exactly by the components of the real moment map parameters $$\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?

$$\mathbb C P^1_i$$ does not make sense universally on $$X_\zeta$$ for all $$\zeta$$. When a parameter $$\zeta$$ cross a wall, the homology class $$[\mathbb CP^1_i]$$ is changed by the Weyl group reflection. Therefore (1) is not correct.
Based on the special case $$G=\mathbb{Z}_2$$, I would guess that formula (1) should be corrected by taking the absolute value $$|\zeta^i_\mathbb{R}|$$ instead of $$\zeta^i_\mathbb{R}$$ itself.
Details of the case $$G=\mathbb{Z}_2$$: In this case $$\zeta_\mathbb{R}$$ is a scalar, and for simplicity of notation let us write $$a$$ for $$\zeta_\mathbb{R}$$. As you remarked, the various hyperkahler quotients $$X_{a,0}(Q_{\mathbb{Z}_2})$$ are birational to $$\mathbb{C}^2/\mathbb{Z}_2$$: the choice of parameter $$a=0$$ for the real moment map corresponds to $$\mathbb{C}^2/\mathbb{Z}_2$$ itself, and the others are its minimal resolution.
You can calculate that the Kähler potential for $$X_{a,0}(Q_{\mathbb{Z}_2})$$, treated as living on $$\mathbb{C}^2/\mathbb{Z}_2$$, is $$\sqrt{a^2+4|{\bf z}|^4} -a\log\left[a+\sqrt{a^2+4|{\bf z}|^4}\right]+a\log |{\bf z}|^2.$$ Sanity check: For $$a=0$$ this gives $$2|{\bf z}|^2$$, thus the Euclidean metric.
The volume of the exceptional sphere is the coefficient of $$\log|{\bf z}|^2$$ for $${\bf z}\to 0$$. For $$a$$ positive, this is $$a$$. However, for $$a$$ negative, \begin{align*} -a\log\left[a+\sqrt{a^2+4|{\bf z}|^4}\right]+a\log |{\bf z}|^2 &=a\log\frac{|{\bf z}|^2}{a+\sqrt{a^2+4|{\bf z}|^4}}\\ &=a\log\frac{|{\bf z}|^2(-a+\sqrt{a^2+4|{\bf z}|^4})}{4|{\bf z}|^4}\\ &=-a\log|{\bf z}|^2 +a\log\left[-a+\sqrt{a^2+4|{\bf z}|^4}\right]+c \end{align*} so the coefficient is $$-a$$.