# Hodge Bundles on Tropical Spaces

I am not sure that this question even makes sense, which I suppose is part of the questions itself.

In any case, I attended a talk recently wherin there was some discussion about a "tropical Teichmuller space", as for example in this paper, associated to the tropical moduli space of abelian varieties. My understanding is that the moduli space of tropical abelian varieties $A_g^{trop}$ can be identified with the skeleton of a Berkovich analytic space associated to the moduli space of abelian varieties $A_g$. Does the Hodge Bundle on the moduli space of abelian varieties "carry over" (whatever that should mean) to the analytic space? If so, can you then "pull it back" to the skeleton, onto which the analytic space deformation retracts?

My question is motivated by the fact that the speaker suggested that this "tropical Teichmuller space" gives some tropical analogy to the Siegel upper half space model for $A_g$, where you take a quotient by $GL_n$ to get the moduli space of tropical abelian varieties. I am basically wondering if there is some notion of a "tropical Siegel modular form", which arises as a section of a "tropical Hodge bundle".

Some nice references for tropical geometry and Berkovich analytic spaces is also welcome.

• This is a very nice question. It seems quite appropriate for Math Overflow, if you don't get a satisfactory answer here. – Alex Wertheim Nov 8 '14 at 23:28
• @AWertheim Thanks! I was planning on waiting for a day or two before going up to MathOverflow. – John Martin Nov 8 '14 at 23:29
• @JohnMartin Note that instead of reposting it there, usual practice is to ask a moderator to move it (so that effort doesn't end up being duplicated on one site after it's answered on another). – user98602 Nov 9 '14 at 20:39
• @MikeMiller I sent a message to the help center asking for help with this, but receieved no response. I just posted it again on MathOverflow with a hyperlink to this post. – John Martin Nov 11 '14 at 2:57