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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?

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# Weight filtration and Hodge theory for tropical varieties

Many concepts is algebraic geometry have tropical analogues. Is there an analogue of the weight filtration or Hodge filtration for tropical varieties?