  In this work we consider generic losses of rank for complex valued matrix
functions depending on two parameters. We give theoretical results that
characterize parameter regions where these losses of rank occur. Our main
results consist in showing how following an appropriate smooth SVD along a
closed loop it is possible to monitor the Berry phases accrued by the singular
vectors to decide if -- inside the loop -- there are parameter values where a
loss of rank takes place. It will be needed to use a new construction of a
smooth SVD, which we call the "joint-MVD" (minimum variation decomposition).
