  [Note: important Corrigendum now available at arXiv:1601.04790] The
isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the
Malgrange's divisor of generalized monodromy data for which a vector bundle is
nontrivial, or, which is the same, a certain Riemann-Hilbert problem has no
solution. In their original work, Jimbo, Miwa, Ueno did not derive the
dependence on the (generalized) monodromy data (i.e. monodromy representation
and Stokes' parameters). We fill the gap by providing a (simpler and more
general) description in which all the parameters of the problem
(monodromy-changing and monodromy-preserving) are dealt with at the same level.
We thus provide variational formulae for the isomonodromic tau function with
respect to the (generalized) monodromy data. The construction applies more
generally: given any (sufficiently well-behaved) family of Riemann-Hilbert
problems (RHP) where the jump matrices depend arbitrarily on deformation
parameters, we can construct a one-form Omega (not necessarily closed) on the
deformation space (Malgrange's differential), defined off Malgrange's divisor.
We then introduce the notion of discrete Schlesinger transformation: it means
that we allow the solution of the RHP to have poles (or zeros) at prescribed
point(s). Even if Omega is not closed, its difference evaluated along the
original solution and the transformed one, is shown to be the logarithmic
differential (on the deformation space) of a function. As a function of the
position of the points of the Schlesinger transformation, yields a natural
generalization of Sato formula for the Baker-Akhiezer vector even in the
absence of a tau function, and it realizes the solution of the RHP as such BA
vector. Some exemples (Painleve' II and finite Toplitz/Hankel determinants) are
provided.
