On ratios of Chern numbers for complex hyperbolic branched covers

TL;DR

This paper investigates the ratios of Chern numbers in complex hyperbolic branched covers, revealing they differ from those of the base manifold in even dimensions, impacting geometric structures and answering a specific open question.

Contribution

It demonstrates that Chern number ratios for complex hyperbolic branched covers differ from the base manifold in even dimensions, addressing a question by Deraux and Seshadri.

Findings

Chern number ratios are not all equal in branched covers

The result applies specifically to even complex dimensions

Implications for the Kähler property of certain metrics

Abstract

In this paper we prove that, at least in even complex dimensions, the ratio of Chern numbers for a closed complex hyperbolic branched cover manifold are not all equal to the corresponding ratio of Chern numbers for a closed complex hyperbolic manifold. This leads to an answer for a question posed by Deraux and Seshadri, and proves that an almost $1/4$-pinched metric constructed by the author in a previous article is not K\"{a}hler.