Kazhdan--Lusztig polynomials of matroids need not be unimodal

We construct, over every finite field, representable matroids whose Kazhdan--Lusztig polynomials are not unimodal. In particular, the conjectures that all Kazhdan--Lusztig polynomials of matroids are log-concave and that they are real-rooted are both false. Our examples are obtained by deleting points from finite projective geometries. More generally, we prove that, under a half-rank degree condition in each contraction quotient, the Kazhdan--Lusztig polynomial of every contraction enumerates the subspaces whose projective points lie in the corresponding deleted set, while the $Z$-polynomial agrees with that of the full projective geometry.