  By enforcing invariance under S-duality in type IIB string theory
compactified on a Calabi-Yau threefold, we derive modular properties of the
generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA
string theory compactified on the same space. Mathematically, these BPS
degeneracies are the generalized Donaldson-Thomas invariants counting coherent
sheaves with support on a divisor $\cal D$, at the large volume attractor
point. For $\cal D$ irreducible, this function is closely related to the
elliptic genus of the superconformal field theory obtained by wrapping M5-brane
on $\cal D$ and is therefore known to be modular. Instead, when $\cal D$ is the
sum of $n$ irreducible divisors ${\cal D}_i$, we show that the generating
function acquires a modular anomaly. We characterize this anomaly for arbitrary
$n$ by providing an explicit expression for a non-holomorphic modular
completion in terms of generalized error functions. As a result, the generating
function turns out to be a (mixed) mock modular form of depth $n-1$.
