We introduce the conormal fan of a matroid <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is a Lagrangian analog of the Bergman fan of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We use the conormal fan to give a Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This allows us to express the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vector of the broken circuit complex of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the intersection theory of the conormal fan of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations. The Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, when combined with the Hodge–Riemann relations for the conormal fan of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, implies Brylawski’s and Dawson’s conjectures that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors of the broken circuit complex and the independence complex of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are log-concave sequences.