Tropical Homology
  Given a tropical variety X and two non-negative integers p and q we define homology group $H_{p,q}(X)$. We show that if X is a smooth tropical variety that can be represented as the tropical limit of a 1-parameter family of complex projective varieties, then $\dim H_{p,q}(X)$ coincides with the Hodge number $h^{p,q}$ of a general member of the family. 
Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, Ilia Zharkov
math.AG