Many interesting questions in four-manifold topology can be phrased in terms of how cohomology classes interact with the geometry of a Riemannian metric. On a closed, oriented four-manifold, harmonic 2-forms split into self-dual and anti-self-dual (ASD) parts, and this splitting varies with the metric. A basic problem is to understand when a given cohomology class, or a collection of classes, can be made anti-self-dual for a suitable choice of metric; this is largely open in general.

In this talk, I will explain a new existence theorem for ASD representatives of geometrically simple classes. More precisely, consider a closed, oriented four-manifold and suppose it contains a collection of pairwise disjoint embedded spheres of small self-intersection. Then there exists a metric for which the cohomology classes of these spheres are all represented by ASD harmonic forms.

The proof combines analytic tools (neck-stretching estimates for elliptic operators) with geometric (Eliashberg's h-principle). I will outline how these ingredients fit together. Time permitting, we also discuss connections with classical period maps from algebraic geometry (e.g., for K3 surfaces).