Polystable Log Calabi-Yau Varieties and Gravitational Instantons

J. Math. Sci. Univ. Tokyo
Vol. 29 (2022), No. 1, Page 1-50.

Odaka, Yuji
Polystable Log Calabi-Yau Varieties and Gravitational Instantons
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Abstract:
Open Calabi-Yau manifolds and log Calabi-Yau varieties have been broadly studied over decades. Regarding them as “semistable” objects, we propose to consider their good proper subclass, which we regard as certain polystable ones, morally corresponding to semistable with closed (minimal) orbits as the classical analogue of GIT.
We partially confirm that the new polystability seems equivalent to the existence of non-compact complete Ricci-flat Kähler metrics with small volume growths, notably many examples of gravitational instantons. Also, we prove some compactness or polystable reduction type results, partially motivated by bubbles of compact Ricci-flat metrics.

Keywords: K-stability, log Calabi-Yau varieties, Ricci-flat Kähler metrics, gravitational instantons.

Mathematics Subject Classification (2010): 14J32, 14L24, 32Q20.
Received: 2020-10-20