  We establish a direct connection between the power of a unitary map in
$d$-dimensions ($d<\infty$) to generate quantum coherence and the geometry of
the set ${\cal M}_d$ of maximally abelian subalgebras (of the quantum system
full operator algebra). This set can be seen as a topologically non-trivial
subset of the Grassmannian over linear operators. The natural distance over the
Grassmannian induces a metric structure on ${\cal M}_d$ which quantifies the
lack of commutativity between the pairs of subalgebras. Given a maximally
abelian subalgebra one can define, on physical grounds, an associated measure
of quantum coherence. We show that the average quantum coherence generated by a
unitary map acting on a uniform ensemble of quantum states in the algebra (the
so-called coherence generating power of the map) is proportional to the
distance between a pair of maximally abelian subalgebras in ${\cal M}_d$
connected by the unitary transformation itself. By embedding the Grassmannian
into a projective space one can pull-back the standard Fubini-Study metric on
${\cal M}_d$ and define in this way novel geometrical measures of quantum
coherence generating power. We also briefly discuss the associated differential
metric structures.
