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Degenerating Kähler–Einstein cones, locally symmetric cusps, and the Tian–Yau metric


Let X be a complex projective manifold and let \(D\subset X\) be a smooth divisor. In this article, we are interested in studying limits when \(\beta \rightarrow 0\) of Kähler–Einstein metrics \(\omega _\beta \) with a cone singularity of angle \(2\pi \beta \) along D. In our first result, we assume that \(X\setminus D\) is a locally symmetric space and we show that \(\omega _\beta \) converges to the locally symmetric metric and further give asymptotics of \(\omega _\beta \) when \(X\setminus D\) is a ball quotient. Our second result deals with the case when X is Fano and D is anticanonical. We prove a folklore conjecture asserting that a rescaled limit of \(\omega _\beta \) is the complete, Ricci flat Tian–Yau metric on \(X\setminus D\). Furthermore, we prove that \((X,\omega _\beta )\) converges to an interval in the Gromov–Hausdorff sense.

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O.B. would like to thank Misha Kapovich for discussions a long time ago about closing complex hyperbolic cusps. H.G. would like to thank Benoît Cadorel for the many insightful discussions about toroidal compactifications of quotients of bounded symmetric domains. The authors would like to thank the referee for reading the manuscript carefully and for the several suggestions that helped us improve the paper. H.G. has benefited from the support of the ANR project GRACK as well as from the state aid managed by the ANR under the ”PIA” program bearing the reference ANR-11-LABX-0040, in connection with the research project HERMETIC.

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Correspondence to Henri Guenancia.

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Biquard, O., Guenancia, H. Degenerating Kähler–Einstein cones, locally symmetric cusps, and the Tian–Yau metric. Invent. math. (2022).

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