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We compute and compare the (intersection) cohomology of various naturalgeometric compactifications of the moduli space of cubic threefolds: the GITcompactification and its Kirwan blowup, as well as the Baily-Borel and toroidalcompactifications of the ball quotient model, due to Allcock-Carlson-Toledo.Our starting point is Kirwan's method. We then follow by investigating thebehavior of the cohomology under the birational maps relating the variousmodels, using the decomposition theorem in different ways, and via a detailedstudy of the boundary of the ball quotient model. As an easy illustration ofour methods, the simpler case of the moduli of cubic surfaces is discussed inan appendix.

Cohomology of the moduli space of cubic threefolds and its smooth models