  We consider the horospherical foliation on any invariant subvariety in the
moduli space of translation surfaces. This foliation can be described
dynamically as the strong unstable foliation for the geodesic flow on the
invariant subvariety, and geometrically, it is induced by the canonical
splitting of $\mathbb{C}$-valued cohomology into its real and imaginary parts.
We define a natural volume form on the leaves of this foliation, and define
horospherical measures as those measures whose conditional measures on leaves
are given by the volume form. We show that the natural measures on invariant
subvarieties, and in particular, the Masur-Veech measures on strata, are
horospherical. We show that these measures are the unique horospherical
measures giving zero mass to the set of surfaces with no horizontal saddle
connections, extending work of Lindenstrauss-Mirzakhani and Hamenstaedt for
principal strata. We describe all the leaf closures for the horospherical
foliation.
