Skip to Main Content
Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces

Authors: Sho Ejiri, Masataka Iwai and Shin-ichi Matsumura; with an appendix by Frédéric Touzet
Journal: J. Algebraic Geom. 32 (2023), 477-517
Published electronically: January 31, 2023
Full-text PDF

Abstract | References | Additional Information

Abstract: In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $\phi \colon X\to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana–Cao–Matsumura’s equality on Hacon–McKernan’s question, whose original proof depends on analytics methods. The third result proves that algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of $Y$. Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.

References [Enhancements On Off] (What's this?)


Additional Information

Sho Ejiri
Affiliation: Department of Mathematics, Graduate School of Science, Osaka Metropolitan University, Osaka City, Osaka 558-8585, Japan
MR Author ID: 1225282

Masataka Iwai
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan
MR Author ID: 1359272
ORCID: 0000-0002-0273-0360

Shin-ichi Matsumura
Affiliation: Mathematical Institute, Tohoku University, 6-3, Aramaki Aza-Aoba, Aoba-ku, Sendai 980-8578, Japan
MR Author ID: 1006398
ORCID: 0000-0002-5928-527X

Frédéric Touzet
ORCID: 0000-0002-0910-8451

Received by editor(s): April 20, 2021
Received by editor(s) in revised form: March 18, 2022
Published electronically: January 31, 2023
Additional Notes: The first author was supported by JSPS KAKENHI Grant $\sharp$18J00171. The second author was supported by JSPS KAKENHI Grant $\sharp$17J04457, by the Program for Leading Graduate Schools MEXT Japan, by the public interest incorporated foundation Fujukai, and by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). The third author was supported by the Grant-in-Aid for Young Scientists (A) $\sharp$17H04821, Grant-in-Aid for Scientific Research (B) $\sharp$ 21H00976, and Fostering Joint International Research (A) $\sharp$19KK0342 from JSPS
Article copyright: © Copyright 2023 University Press, Inc.