  To prove log-concavity of the characteristic polynomials of matroids,
Adiprasito-Huh-Katz proved the K\"{a}hler 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 K\"{a}hler package in the same way as compact K\"{a}hler
manifolds.
