It is well known (see [Derdzinski](https://eudml.org/doc/89617)) that for a Kaehler metric on a four-manifold, its self-dual Weyl curvature has only two distinct eigenvalues:
$$-\frac{R}{12},\ -\frac{R}{12},\ \frac{R}{6}.$$


I was wondering whether anti-self-dual Weyl curvature also has only two distinct eigenvalues (may not be written explicitly in terms of $R$)? 


Or is it true if we restrict to Kaehler-Einstein metrics?



A question on anti-self-dual Weyl curvature of Kaehler surfaces
The answer is 'no'. In fact, even for a Ricci-flat Kähler manifold $(M^4,J,g)$, the map $W\_-(x):\Lambda^2\_-(T\_xM)\to\Lambda^2\_-(T\_xM)$ can be any symmetric traceless linear map. Hence, the only constraint on the eigenvalues is that they sum to zero. 


This fact was known to Élie Cartan already in 1926, although he did not use such terminology, as it had not yet been invented. What he considered was Riemannian 4-manifolds with holonomy contained in $\mathrm{SU}(2)\subset\mathrm{SO}(4)$ (what we now call a Calabi-Yau metric in complex dimension $2$). He observed that such metrics are Ricci-flat and that (with the orientation correctly specified) the curvature of such a metric consists entirely of what we now call $W\_-$, the anti-self dual Weyl curvature, and he showed that $W\_-(x):\Lambda^2\_-(T\_xM)\to\Lambda^2\_-(T\_xM)$ could assume any trace-free symmetric value. (See his little book, *Leçons sur la géométrie des espaces de Riemann*, particularly the last couple of pages, though you'll have to consult earlier sections in order to correctly interpret his terminology.)


Of course, Cartan only considered the local theory, not global examples. However, it is not hard to show that, up to diffeomorphism, the space of germs of $\mathrm{SU}(2)$-holonomy metrics in dimension $4$ for which $W\_-$ has a double eigenvalue everywhere is finite dimensional and that such metrics (on simply connected domains) always have non-trivial Killing vector fields. In particular, the Calabi-Yau metrics on K3 surfaces are not of this kind, hence, their $W\_-$ must have three distinct eigenvalues at a generic point.


