Non–trivial Harmonic Spinors on Generic Algebraic Surfaces

For every closed Riemannian spin manifold its Dirac operator is a selfadjoint linear elliptic operator acting on sections of the spinor bundle. Hitchin [3] proved that the dimension of its kernel, the space of harmonic spinors, only depends on the conformal class of the Riemannian metric, and that it varies with the conformal class. However, the variation is not understood. Investigating this variation is particularly interesting for manifolds of dimension 4k. In that case the Dirac operator interchanges spinors of different chirality and the index of the (half–) Dirac operator D : V+ → V− given by the –genus can be non–zero. We say that a conformal structure admits non–trivial harmonic spinors if the dimension of the space of harmonic spinors is larger than |index(D)|. This is equivalent to the existence of harmonic spinors of both chiralities. The natural problem to consider is: