There are only very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions larger than 24 the known examples are diffeomorphic to locally rank 1 symmetric spaces. I will construct metrics with positive sectional curvature on open and dense sets of points on the projective tangent bundles of RP^n, CP^n and HP^n. The so called deformation conjecture says that these kind of metrics can be deformed into metrics with positive sectional curvature everywhere. However, the simplest new example within our class, the projective tangent bundle of RP^3, is diffeomorphic to the product RP^3 x RP^2. This non-orientable manifold is known not to admit a metric with positive sectional curvature. Thus the construction provides a counterexample to the deformation conjecture.