Cyclic branched covers of complex hyperbolic manifolds have non-complex-hyperbolic Chern number ratios, proven exactly in dimension 2 and claimed with a gap in higher even dimensions.
Goldman, Michael Kapovich, and Bernhard Leeb
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On ratios of Chern numbers for complex hyperbolic branched covers
Cyclic branched covers of complex hyperbolic manifolds have non-complex-hyperbolic Chern number ratios, proven exactly in dimension 2 and claimed with a gap in higher even dimensions.