To prove log-concavity of the characteristic polynomials of matroids, Adiprasito-Huh-Katz proved the Kähler package (the hard Lefschetz theorem and the Hodge-Riemann bilinear relations) for their Chow rings. Amini-Piquerez generalized it to tropical cohomology of smooth projective tropical varieties. Their proofs were combinatorial. In this paper, we establish tropical Hodge theory except for regularity of solutions of Laplacian, and give a conditional proof of the Kähler package in the same way as compact Kähler manifolds.