  We show that a genus $2$ curve over a number field whose jacobian has complex
multiplication will usually have stable bad reduction at some prime. We prove
this by computing the Faltings height of the jacobian in two different ways.
First, we use a formula by Colmez and Obus specific to the CM case and valid
when the CM field is an abelian extension of the rationals. This formula links
the height and the logarithmic derivatives of an $L$-function. The second
formula involves a decomposition of the height into local terms based on a
hyperelliptic model. We use results of Igusa, Liu, and Saito to show that the
contribution at the finite places in our decomposition measures the stable bad
reduction of the curve and subconvexity bounds by Michel and Venkatesh together
with an equidistribution result of Zhang to handle the infinite places.
