Question

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?

Answer 1

I'm not quite sure if there's a useful notion of Hodge/weight filtration on a tropical variety. If we look at tropical varieties that are tropicalization of algebraic varieties over a non-Archimedean field, the topology of the tropical variety is related to the lowest weight bit of the weight filtration. I don't know how that bit is naturally filtered any further.

The question I was curious about in my research statement is: is there a combinatorial way to encode higher bits of the weight filtration? I suspect that they can be expressed as a complex of sheaves on the tropical variety.

I think there's probably a precise way of formulating your statement about tropical curves as "the Abel-Jacobi map commutes with tropicalization for totally degenerate curves." For details, look at p.19 of my paper with David Helm. I'm not sure if there's a natural way to tropicalize the period domain, but that'd be a fun question to address.

Answer 2

Itenberg, Kazarkov, Mikhalkin and Zharkov have formulated a definition of tropical $H^{p,q}$ and presented it at a number of conferences, although I don't think that there is a preprint yet. You can watch Zharkov's talk on this from 2009 here.

Their definition is restricted to the case that $\mathrm{Trop} \ X$ locally looks like a tropical linear space. (For example, if $X$ is a curve then, at a vertex of degree $d$, the directions of the incoming edges must span a space of dimension $d-1$ and the unique relation between the minimal lattice vectors on these directions must be that their sum is $0$.) This can be thought of as a tropical "smoothness" condition.

Roughly speaking, $H^{0,q}$ is related to the topological cohomology of $\mathrm{Trop} \ X$ which should, in this context, be viewed as the cohomology of the sheaf of locally constant functions on $\mathrm{Trop} \ X$. $H^{p,q}(X)$ is related to the cohomology of a sheaf on $\mathrm{Trop} \ X$ which is related to the degree $p$ part of Orlik-Solomon algebras of the matroids locally describing the relevant tropical linear spaces. (In other words, to $H^p$ of the corresponding hyperplane arrangement complement.)

Until a preprint arrives, the best reference seems to be the papers of Kristin Shaw, a student of Mikhalkin. See Section 2.2 of her thesis for a good start.