Polystable log Calabi-Yau varieties and Gravitational instantons

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 poly-stable 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 Kahler 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.

arXiv comment40 pages

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Cited by in corpus (3)

• Degenerating Kähler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric
• PL density invariant for type II degenerating K3 surfaces, Moduli compactification and hyperKahler metrics
• Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures