{"title":"On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces","authors":"Sho Ejiri, M. Iwai, Shin-ichi Matsumura","doi":"10.1090/jag/814","DOIUrl":null,"url":null,"abstract":"In this paper, we study the relative anti-canonical divisor \n\n \n \n −\n \n K\n \n X\n \n /\n \n Y\n \n \n \n -K_{X/Y}\n \n\n of an algebraic fiber space \n\n \n \n ϕ\n :\n X\n →\n Y\n \n \\phi \\colon X\\to Y\n \n\n, and we reveal relations among positivity conditions of \n\n \n \n −\n \n K\n \n X\n \n /\n \n Y\n \n \n \n -K_{X/Y}\n \n\n, 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 \n\n \n Y\n Y\n \n\n. Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2020-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"12","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/814","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 12
Abstract
In this paper, we study the relative anti-canonical divisor
−
K
X
/
Y
-K_{X/Y}
of an algebraic fiber space
ϕ
:
X
→
Y
\phi \colon X\to Y
, and we reveal relations among positivity conditions of
−
K
X
/
Y
-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
Y
. Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.
期刊介绍:
The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology.
This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.