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Abstract We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety. 

We introduce the conormal fan of a matroid M \operatorname {M} , which is a Lagrangian analog of the Bergman fan of M \operatorname {M} . We use the conormal fan to give a Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of M \operatorname {M} . This allows us to express the h h -vector of the broken circuit complex of M \operatorname {M} in terms of the intersection theory of the conormal fan of M \operatorname {M} . 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 M \operatorname {M} , when combined with the Hodge–Riemann relations for the conormal fan of M \operatorname {M} , implies Brylawski’s and Dawson’s conjectures that the h h -vectors of the broken circuit complex and the independence complex of M \operatorname {M} are log-concave sequences.