Explicit Gromov–Hausdorff compactifications of moduli spaces of Kähler–Einstein Fano manifolds

Song Sun (Department of Mathematics, University of California, Berkeley, Ca., U.S.A.; and Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Abstract

We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler–Einstein Fano manifolds in all complex dimensions bigger than two (Fano $\mathrm{K}$-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of $\mathrm{K}$-stable Fano manifolds with large anti-canonical volume. Our arguments are based on recent progress about the geometry of metric tangent cones and on related ideas about the algebro-geometric study of singularities of $\mathrm{K}$-stable Fano varieties.

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Dedicated to Sir Simon Donaldson on his 60th birthday.

C.S. is partially supported by AUFF Starting Grant 24285.

S.S. is partially supported by NSF grant DMS-1405832 and an Alfred P. Sloan Fellowship.

Received 29 May 2017

Published 12 November 2018