  Probing the magnetism of the upper solar chromosphere requires measuring and
modeling the scattering polarization produced by anisotropic radiation pumping
in UV spectral lines. Here we apply PORTA (a novel radiative transfer code) to
investigate the hydrogen Ly$\alpha$ line in a 3D model of the solar atmosphere
resulting from a state of the art MHD simulation. At full spatial resolution
the linear polarization signals are very significant all over the solar disk,
with a large fraction of the field of view showing line-center amplitudes well
above the 1% level. Via the Hanle effect the line-center polarization signals
are sensitive to the magnetic field of the model's transition region, even when
its mean field strength is only 15 G. The breaking of the axial symmetry of the
radiation field produces significant forward-scattering polarization in
Ly$\alpha$, without the need of an inclined magnetic field. Interestingly, the
Hanle effect tends to decrease such forward-scattering polarization signals in
most of the points of the field of view. When the spatial resolution is
degraded, the line-center polarization of Ly$\alpha$ drops below the 1% level,
reaching values similar to those previously found in 1D semi-empirical models
(i.e., up to about 0.5%). The center to limb variation of the
spatially-averaged polarization signals is qualitatively similar to that found
in 1D models, with the largest line-center amplitudes at $\mu=\cos\theta\approx
0.4$ ($\theta$ being the heliocentric angle). These results are important, both
for designing the needed space-based instrumentation and for a reliable
interpretation of future observations of the Ly$\alpha$ polarization.
