  Let (M,g) be a compact oriented Einstein 4-manifold. Write R-plus for the
part of the curvature operator of g which acts on self-dual 2-forms. We prove
that if R-plus is negative definite then g is locally rigid: any other Einstein
metric near to g is isometric to it. This is a chiral generalisation of Koiso's
Theorem, which proves local rigidity of Einstein metrics with negative
sectional curvatures. Our hypotheses are roughly one half of Koiso's. Our proof
uses a new variational description of Einstein 4-manifolds, as critical points
of the so-called poure connection action S. The key step in the proof is that
when R-plus is negative definite, the Hessian of S is strictly positive modulo
gauge.
