  The previously proposed "Complexity=Volume" or CV-duality is probed and
developed in several directions. We show that the apparent lack of universality
for large and small black holes is removed if the volume is measured in units
of the maximal time from the horizon to the "final slice" (times Planck area).
This also works for spinning black holes. We make use of the conserved "volume
current", associated with a foliation of spacetime by maximal volume surfaces,
whose flux measures their volume. This flux picture suggests that there is a
transfer of the complexity from the UV to the IR in holographic CFTs, which is
reminiscent of thermalization behavior deduced using holography. It also
naturally gives a second law for the complexity when applied at a black hole
horizon. We further establish a result supporting the conjecture that a
boundary foliation determines a bulk maximal foliation without gaps, establish
a global inequality on maximal volumes that can be used to deduce the
monotonicity of the complexification rate on a boost-invariant background, and
probe CV duality in the settings of multiple quenches, spinning black holes,
and Rindler-AdS.
