  We study two recent conjectures for holographic complexity: the
complexity=action conjecture and the complexity=volume conjecture. In
particular, we examine the structure of the UV divergences appearing in these
quantities, and show that the coefficients can be written as local integrals of
geometric quantities in the boundary. We also consider extending these
conjectures to evaluate the complexity of the mixed state produced by reducing
the pure global state to a specific subregion of the boundary time slice. The
UV divergences in this subregion complexity have a similar geometric structure,
but there are also new divergences associated with the geometry of the surface
enclosing the boundary region of interest. We discuss possible implications
arising from the geometric nature of these UV divergences.
