Complexity change under conformal transformations in $AdS_{3}/CFT_{2}$

Using the volume proposal, we compute the change of complexity of holographic states caused by a small conformal transformation in $AdS_{3}/CFT_{2}$. This computation is done perturbatively to second order. We give a general result and discuss some of its properties. As operators generating such conformal transformations can be explicitly constructed in CFT terms, these results allow for a comparison between holographic methods of defining and computing computational complexity and purely field-theoretic proposals. A comparison of our results to one such proposal is given.