On ratios of Chern numbers for complex hyperbolic branched covers
Barry Minemyer
TL;DR
The paper investigates ratios of Chern numbers for complex hyperbolic branched covers of even complex dimensions and shows these ratios are not all equal to the corresponding ratios of a closed complex hyperbolic manifold. It combines Hirzebruch proportionality with signature and ramified-covering formulas to derive explicit expressions for the signature and Chern numbers of the branched cover $X$, expressed in terms of data from the base pair $(M,N)$ and the branching parameters $(d,m)$. In dimension two, a concrete formula $c_1^2(X)-3c_2(X)= m \frac{(d-1)^2}{2d}\chi(N)$ is obtained, which is nonzero for all $d\ge2$, proving that $X$ cannot be complex hyperbolic and that not all Chern-number ratios match those of $\bC P^n$. For higher even dimensions, the paper shows that, for all but finitely many $d$, at least one Chern-number ratio diverges from the complex-hyperbolic ratio; this yields a negative answer to a Deraux–Seshadri question and implies the constructed almost $1/4$-pinched metrics on these branched covers are not Kähler. The results hinge on explicit ramified-covering signatures and Chern-class calculations, highlighting the a priori rigid relationship between curvature pinching, topology, and complex structure in this setting.
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 Kähler.