The Riemann-Roch theorem for tropical curves was shown by Gathmann-Kerber and Mikhalkin-Zharkov in 2008. It is a very interesting problem to generalize the tropical Riemann-Roch theorem to higher dimensions, while there are few results for this problem. A main obstacle to higher dimensional generalization is to define the Euler characteristic of a tropical line bundle since the heigher sheaf cohomology cannot be defined as ordinary way.
In this talk, we study the space of global sections of line bundles over tropical tori, called tropical theta functions, and show the Riemann-Roch inequality for tropical abelian surfaces and more results.