  We analyze isomonodromic deformations of rational connections on the Riemann
sphere with Fuchsian and irregular singularities. The Fuchsian singularities
are allowed to be of arbitrary resonant index; the irregular singularities are
also allowed to be resonant in the sense that the leading coefficient matrix at
each singularity may have arbitrary Jordan canonical form, with a genericity
condition on the Lidskii submatrix of the subleading term. We also give the
relevant notion of isomonodromic tau function extending the one of non-resonant
deformations introduced by Miwa-Jimbo-Ueno. The tau function is expressed
purely in terms of spectral invariants of the matrix of the connection.
