It is well known (see Derdzinski) 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?


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.

  • $\begingroup$ Thank you very much for your answer. If we further restrict to Fano Kaehler-Einstein metric, may it be true? (They have been classified by Gang Tian, but I couldn't find any paper describing $W^-$ of these manifolds) $\endgroup$ – littlelittlelittle Dec 29 '15 at 7:15
  • $\begingroup$ It is certainly not true in general. The space of isometry classes of Kähler-Einstein metrics on complex surfaces that are Fano (i.e., have positive Einstein constant) and have the property that $W_-$ has a double eigenvalue at every point is only three dimensional, and this is not large enough to account for all the Fano surfaces, or even the nonsingular cubic surfaces in $\mathbb{P}^3$. $\endgroup$ – Robert Bryant Dec 30 '15 at 14:09

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