Anti-self-dual 4-manifolds and Kähler geometry

An oriented Riemannian 4-manifold $(M,g)$ is said to be anti-self-dual if its Weyl curvature satisfies $W=-*W$, where $*$ is the Hodge star operator acting on bundle-valued 2-forms. Since the Weyl curvature is precisely the part of the Riemann curvature tensor which is invaraint under rescalings $g\mapsto e^ug$, this condition is conformally invariant: if $g$ is anti-self-dual, so is every metric in its conformal class $[g]=\{e^ug\}$.

If $(M,g,J)$ is a Kähler manifold of real dimension 4, then $(M,g)$ is anti-self-dual iff its scalar curvature vanishes [4]. In conjunction with the Penrose twistor correspondence [10, 1], this unexpected link between conformal geometry and Kähler geometry allows one to prove the following:
Theorem 1 [8, 5] Let $(M,J)$ be a compact complex surface which admits a Kähler metric for which the integral of the scalar curvature is non-negative. Then precisely one of the following holds: $(M,J)$ admits a Ricci-flat Kähler metric; or any blow-up of $(M,J)$ has blow-ups which admit scalar-flat Kähler metrics.

A key feature of the proof is that a large number of the scalar-flat Kähler surfaces in question can be written down explicitly [6, 7]; others are obtained from explicit orbifold solutions by a smoothing procedure [9, 5]. This concrete aspect of the proof allows one to estimate the number of blow-ups necessary in particular cases; for example, suitable 14-point blow-ups of ${\bf C}{\rm P}_2$ admit scalar-flat Kähler metrics. Used with a theorem of Donaldson-Friedman/Floer [2, 3], this allows one to conclude the following:
Theorem 2 [9] There are anti-self-dual metrics on $m{\bf C}{\rm P}_2\#n\overline{{\bf C}{\rm P}}_2$ if $n\geq 14m$. \par\noindent {\bf Corollary 1} Let $M$ be a simply-connected compact oriented smooth 4-manifold. Then a connected sum of $M$ with sufficiently many ${\bf C}{\rm P}_2$'s and $\overline{{\bf C}{\rm P}}_2$'s admits anti-self-dual metrics.

Taubes [11] has proved a more powerful result: one can always find anti-self-dual metrics on the connected sum of a smooth oriented 4-manifold $M$ with enough copies of $\overline{{\bf C}{\rm P}}_2$. On the other hand, the techniques described here yield more refined information in many concrete cases.

Bibliography:

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