Abstract
We study the degenerations of asymptotically conical Ricci-flat Kähler metrics as the Kähler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kähler metrics converge to a incomplete smooth Ricci-flat Kähler metric away from a compact subvariety. As a consequence, we construct singular Calabi–Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our results to study several classes of examples of geometric transitions between Calabi–Yau manifolds.
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Communicated by Ngaiming Mok.
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T. C. Collins is supported in part by NSF grant DMS-1944952, DMS-1810924 and an Alfred P. Sloan Fellowship. B. Guo is supported in part by NSF grant DMS-1945869. F. Tong is supported in part by NSF grant DMS-1855947.
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Collins, T.C., Guo, B. & Tong, F. On the degeneration of asymptotically conical Calabi–Yau metrics. Math. Ann. 383, 867–919 (2022). https://doi.org/10.1007/s00208-021-02229-z
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DOI: https://doi.org/10.1007/s00208-021-02229-z