Betti number
In dimension 3, there are smooth complex Fano varieties which are not rational, for example cubic 3-folds in P4 (by Clemens - Griffiths) and quartic 3-folds in P4 (by Iskovskikh - Manin). classified the smooth Fano 3-folds with second Betti number 1 into 17 classes, and classified the smooth ones with second Betti number at least 2, finding 88 deformation classes. A detailed summary of the classification of smooth Fano 3-folds is given in .