Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 2

Special Issue in Honor of Yuri Manin: Part 2 of 2

Guest Editors: Lizhen Ji, Kefeng Liu, Yuri Tschinkel, and Shing-Tung Yau

Complete complexes and spectral sequences

Pages: 215 – 246

DOI: https://dx.doi.org/10.4310/PAMQ.2017.v13.n2.a2


Mikhail Kapranov (Kavli IPMU, UTIAS, University of Tokyo, Kashiwa, Chiba, Japan)

Evangelos Routis (Kavli IPMU, UTIAS, University of Tokyo, Kashiwa, Chiba, Japan)


By analogy with the classical (Chasles–Schubert–Semple–Tyrell) spaces of complete quadrics and complete collineations, we introduce the variety of complete complexes. Its points can be seen as equivalence classes of spectral sequences of a certain type. We prove that the set of such equivalence classes has a structure of a smooth projective variety. We show that it provides a desingularization, with normal crossings boundary, of the Buchsbaum–Eisenbud variety of complexes, i.e., a compactification of the union of its maximal strata.

Received 29 September 2017

Published 14 September 2018