On Ahlfors currents

TL;DR

This paper demonstrates that Ahlfors currents associated with the same entire curve can be nonunique, constructing examples with diverse properties and providing new instances on complex surfaces.

Contribution

It constructs explicit examples of entire curves producing infinitely many cohomologically distinct Ahlfors currents, revealing nonuniqueness and complex structure in their decomposition.

Findings

Ahlfors currents can be nonunique for the same entire curve.

Constructed examples include currents with singular and diffuse parts.

New examples of diffuse Ahlfors currents on complex surfaces.

Abstract

We answer a basic question in Nevanlinna theory that Ahlfors currents associated to the same entire curve may be nonunique. Indeed, we will construct one exotic entire curve $f: \mathbb{C}\rightarrow X$ which produces infinitely many cohomologically different Ahlfors currents. Moreover, concerning Siu's decomposition, for an arbitrary $k\in \mathbb{Z}_{+}\cup \{\infty\}$, some of the obtained Ahlfors currents have singular parts supported on $k$ irreducible curves. In addition, they can have nonzero diffuse parts as well. Lastly, we provide new examples of diffuse Ahlfors currents on the product of two elliptic curves and on $\mathbb{P}^2(\mathbb{C})$, and we show cohomologically elaborate Ahlfors currents on blow-ups of $X$.