K-theory, reality, and duality

We show that the real K-theory spectrum KO is Anderson self-dual using the method previously employed in the second author's calculation of the Anderson dual of Tmf. Indeed the current work can be considered as a lower chromatic version of that calculation. Emphasis is given to an algebro-geometric interpretation of this result in spectrally derived algebraic geometry. We finish by applying the result to a calculation of 2-primary Gross-Hopkins duality at height 1, and obtain an independent calculation of the group of exotic elements of the K(1)-local Picard group.