  We introduce the intersection cohomology module of a matroid and prove that
it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the
Hodge-Riemann relations. As applications, we obtain proofs of Dowling and
Wilson's Top-Heavy conjecture and the nonnegativity of the coefficients of
Kazhdan-Lusztig polynomials for all matroids.
