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In a recent paper, the first author proved the log-concavity of thecoefficients of the characteristic polynomial of a matroid realizable over afield of characteristic 0, answering a long-standing conjecture of Read ingraph theory. We extend the proof to all realizable matroids, making progresstowards a more general conjecture of Rota-Heron-Welsh. Our proof follows froman identification of the coefficients of the reduced characteristic polynomialas answers to particular intersection problems on a toric variety. Thelog-concavity then follows from an inequality of Hodge type.

Log-concavity of characteristic polynomials and the Bergman fan of matroids