Chapter 5: Tropical holomorphic curves

Cartan's generalization of Nevanlinna's second main theorem for holomorphic curves in the finite dimensional projective space is an important extension of value distribution theory, which has surprising applications in the study of meromorphic functions in the complex plane as well. The purpose of this chapter is to introduce a tropical version of Cartan's value distribution theory. Following [67], we will define a tropical analogue of Cartan's characteristic function, and show that it is well defined and reduces exactly to the tropical characteristic function introduced in Chapter 3 in the special case where the tropical holomorphic curve is one-dimensional. The central result of this section is a tropical analogue of Cartan's second theorem. We will show that this result is a natural extension of the tropical second main theorem, Theorem 3.40, by proving that under a certain natural non-degeneracy condition on targets, the tropical Cartan theorem reduces to Theorem 3.40 in the one-dimensional case. The tropical Cartan's second main theorem also implies a second main theorem type inequality, which appears to include a previously unknown ramification type term. We will conclude the section by a discussion on ramification in the tropical meromorphic functions. In order to study the properties of tropical holomorphic curves, we first need to introduce some notions from tropical linear algebra.