  In this paper we propose a new class of preconditioners for the isogeometric
discretization of the Stokes system. Their application involves the solution of
a Sylvester-like equation, which can be done efficiently thanks to the Fast
Diagonalization method. These preconditioners are robust with respect to both
the spline degree and mesh size. By incorporating information on the geometry
parametrization and equation coefficients, we maintain efficiency on
non-trivial computational domains and for variable kinematic viscosity. In our
numerical tests we compare to a standard approach, showing that the overall
iterative solver based on our preconditioners is significantly faster.
