  The holographic complexity is UV divergent. As a finite complexity, we
propose a "regularized complexity" by employing a similar method to the
holographic renormalization. We add codimension-two boundary counterterms which
do not contain any boundary stress tensor information. It means that we
subtract only non-dynamic background and all the dynamic information of
holographic complexity is contained in the regularized part. After showing the
general counterterms for both CA and CV conjectures in holographic spacetime
dimension 5 and less, we give concrete examples: the BTZ black holes and the
four and five dimensional Schwarzschild AdS black holes. We propose how to
obtain the counterterms in higher spacetime dimensions and show explicit
formulas only for some special cases with enough symmetries. We also compute
the complexity of formation by using the regularized complexity.

High Energy Physics - Theory