The anti-self-dual deformation complex and a conjecture of Singer

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>M</m:mi> <m:mn>4</m:mn> </m:msup> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M⁴,g) be a smooth, closed, oriented anti-self-dual (ASD) four-manifold. <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>M</m:mi> <m:mn>4</m:mn> </m:msup> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M⁴,g) is said to be unobstructed if the cokernel of the linearisation of the self-dual Weyl tensor is trivial. This condition can also be characterised as the vanishing of the second cohomology group of the ASD deformation complex, and is central to understanding the local structure of the moduli space of ASD conformal structures. It also arises in construction of ASD manifolds by twistor and gluing methods. In this article, we give conformally invariant conditions which imply an ASD manifold of positive Yamabe type is unobstructed.