  The Bialynicki-Birula decomposition of the space of lambda-connections
restricts to the Morse stratification on the moduli space of Higgs bundles and
to the partial oper stratification on the de Rham moduli space of holomorphic
connections. For both the Morse and partial oper stratifications, every stratum
is a holomorphic Lagrangian fibration over a component of the space of complex
variations of Hodge structure. In this paper, we generalize known results for
the Hitchin section and the space of opers to arbitrary strata. These include
the following: a biholomorphic identification of the fibers of the two strata
over a stable variation of Hodge structure via the "h-bar-conformal limit" of
Gaiotto, a proof that the fibers of the Morse and partial oper stratifications
are transverse at the base point, and an explicit parametrization of the fibers
as half-dimensional affine spaces.
