Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 1

Special Issue in Honor of Yuri Manin: Part 3 of 3

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

Skrepa morphisms

Pages: 35 – 124

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n1.a2


V. V. Shokurov (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.; Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia; and Laboratory AGHA, Moscow Institute of Physics and Technology, Moscow, Russia)


For a complete algebraic space with a nef quasipolarization and its closed subspace which includes the exceptional locus of the quasipolarization, the paper establishes the existence of an infinitesimal neighborhood of the subspace such that for any quasipolarized morphism of the neighborhood into a polarized scheme there exists a quasipolarized modification of the space into a polarized scheme. This is a quasipolarized version of the wellknown Artin modification. As applications nonprojective versions of the semiampleness criterion of Birkar in general and of Keel in positive characteristics are obtained.

Partially supported by NSF grant DMS-1400943.

Received 12 June 2017

Published 6 February 2020