Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 3

Special Issue in Honor of Simon Donaldson

Guest Editors: Kefeng Liu, Richard Thomas, and Shing-Tung Yau

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

Pages: 477 – 515

DOI: https://dx.doi.org/10.4310/PAMQ.2017.v13.n3.a5


Cristiano Spotti (Centre for Quantum Geometry of Moduli Spaces, Aarhus University, Aarhus, Denmark)

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.)


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.

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