  The gravitino mass is expected not to be much smaller than the Planck scale
for a large fraction of vacua in flux compactifications. There is no continuous
parameter to tune even by hand, and it seems that the gravitino mass can be
small only as a result of accidental cancellation among period integrals
weighted by integer-valued flux quanta. DeWolfe et.al. (2005) proposed to pay
close attention to vacua where the Hodge decomposition is possible within a
number field, so that the precise cancellation takes place as a result of
algebra. We focus on a subclass of those vacua---those with complex
multiplications---and explore more on the idea in this article. It turns out,
in Type IIB compactifications, that those vacua admit non-trivial
supersymmetric flux configurations if and only if the reflex field of the Weil
intermediate Jacobian is isomorphic to the quadratic imaginary field generated
by the axidilaton vacuum expectation value. We also found that flux statistics
is highly enriched on such vacua, as F-term conditions become linearly
dependent.
