A factorisation theorem for curve with vanishing self-intersection

TL;DR

This paper investigates conditions under which a curve with zero self-intersection in a compact Kähler surface is a fiber of a holomorphic map, using solutions to a Poincaré-Lelong equation on infinite coverings.

Contribution

It provides partial positive answers to when such curves are fibers of holomorphic maps, employing a novel approach via solving a Poincaré-Lelong equation.

Findings

Under certain conditions, curves with vanishing self-intersection are fibers of holomorphic maps.

The method involves solving a Poincaré-Lelong equation on infinite coverings.

Results contribute to understanding the structure of curves in Kähler surfaces.

Abstract

Let C be a curve in a compact Kahler surface Y. Assume that the self-intersection of C is vanishing, and the image of the fundamental group of C in Y is of infinite index in the fundamental group of Y. Does it follow that C is a fiber of a holomorphic map from Y to a curve ? We give, in some cases, a positive answer to the question. Our strategy is to solve a Poincare-Lelong equation on some infinite covering of Y.