  This paper is the first in a series devoted to the development of a Hodge
theory for tropical varieties. We introduce a notion of T-stability for
tropical fans and prove that various geometric properties of tropical fans are
T-stable. As a consequence, we establish K\"ahler properties for the Chow ring
in a large class of tropical fans, going beyond the case of matroids and their
Bergman fans. As a by-product, we obtain a new proof of the K\"ahler package
for combinatorial geometries. The approach makes it possible to deal with
tropical fans with general weights.
