For the Eisenstein Deligne-Mostow variety linked to 12 points on P^1, Kirwan's partial resolution is not semi-toroidal, the period map does not lift to the toroidal compactification, the two compactifications are not derived equivalent, and an automorphic form is constructed on non-hyperelliptic g=4
Koll´ ar,Singularities of the minimal model program , With the collaboration of S´ andor Kov´ acs., Cambridge Tracts in Mathematics 200 (2013)
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Compactifications of the Eisenstein ancestral Deligne-Mostow variety
For the Eisenstein Deligne-Mostow variety linked to 12 points on P^1, Kirwan's partial resolution is not semi-toroidal, the period map does not lift to the toroidal compactification, the two compactifications are not derived equivalent, and an automorphic form is constructed on non-hyperelliptic g=4