Kronheimer's results on ALE spaces as hyperkahler quotients

Question

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?

Answer 1

$\mathbb C P^1_i$ 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.

Answer 2

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 $-a$.