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: http://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.

Full Text (PDF format)

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