Many concepts is algebraic geometry have tropical analogues.
Question: Is there an analogue of the weight filtration or Hodge filtration for tropical varieties?
A tropical curve ends up being essentially a metric graph. The tropical genus is the first Betti number of the graph. There is a period mapping (analogous to the classical Abel-Jacobi period map) from the moduli space of tropical curves of genus $g$ to the space $GL_g(\mathbb{R})/O_g(\mathbb{R})$. Can this period map be interpreted as a classifying map for variation of tropical Hodge structure?
