  We consider subregion complexity within the AdS3/CFT2 correspondence. We
rewrite the volume proposal, according to which the complexity of a reduced
density matrix is given by the spacetime volume contained inside the associated
Ryu-Takayanagi (RT) surface, in terms of an integral over the curvature. Using
the Gauss-Bonnet theorem we evaluate this quantity for general entangling
regions and temperature. In particular, we find that the discontinuity that
occurs under a change in the RT surface is given by a fixed topological
contribution, independent of the temperature or details of the entangling
region. We offer a definition and interpretation of subregion complexity in the
context of tensor networks, and show numerically that it reproduces the
qualitative features of the holographic computation in the case of a random
tensor network using its relation to the Ising model. Finally, we give a
prescription for computing subregion complexity directly in CFT using the
kinematic space formalism, and use it to reproduce some of our explicit gravity
results obtained at zero temperature. We thus obtain a concrete matching of
results for subregion complexity between the gravity and tensor network
approaches, as well as a CFT prescription.
