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We completely determine the intersection cohomology of the Satake compactifications \({{\mathcal {A}}_{2}^{\mathrm{Sat}}},{{\mathcal {A}}_{3}^{\mathrm{Sat}}}\), and \({{\mathcal {A}}_{4}^{\mathrm{Sat}}}\), except for \({ IH}^{10}({{\mathcal {A}}_{4}^{\mathrm{Sat}}})\). We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved.

The intersection cohomology of the Satake compactification of $${\mathcal {A}}_g$$ for $$g \le 4$$