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Ｆ純閾値の昇鎖条件について

关于F纯阈值的递增链条件

基本信息

批准号：
17J04317
负责人：
佐藤謙太
金额：
$ 1.09万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for JSPS Fellows
财政年份：
2017
资助国家：
日本
起止时间：
2017-04-26 至 2019-03-31
项目状态：
已结题

项目摘要

今年度も前年度に引き続き，正標数の代数多様体におけるF純閾値の昇鎖条件について研究を行った．F純閾値とは，正標数の代数多様体とその上のイデアル層の対に対して定まる非負実数で，その対の特異点の悪さを反映する量である．以下，研究成果を記述する．前年度の研究において，次のような結果を示していた：鋭F純特異点しか持たない多様体を，埋め込み次元を固定しながら動かし，その上のイデアル層も自由に動かす時に現れうるF純閾値の全体のなす集合は昇鎖条件を満たす．今年度はまず，この結果を応用することで，標数0の対数的標準閾値の昇鎖条件にまつわるde Fernex-Ein-Mustataの結果を正標数化することに成功した．すなわち，多様体が局所完全交差特異点しか持たない場合に限定すれば，埋め込み次元でなく(クルル)次元を固定して多様体を動かした場合にもF純閾値について同様の昇鎖条件が満たされる，ということを証明した．ここまでの結果をプレプリントとして発表した．今年度は更に，F純閾値の昇鎖条件の応用として，大域的F正則多様体の有界性にアプローチできないかについて研究を行った．Hacon-McKernan-Xuらは，標数0の対数的標準閾値の昇鎖条件を証明する過程で，Gorenstein 指数と次元を固定したFano多様体の反標準体積の一様有界性を示していた．その議論を正標数化することにより，後述の予想1と予想2を仮定することで，大域的F正則な非特異Fano多様体の反標準体積の一様有界性が従うことを確認した．ここで，予想1はF純閾値の昇鎖条件が，環の埋め込み次元の代わりにクルル次元を固定した状態でも成立する，という予想であり，予想2は，大域的F正則なFano多様体X上で，反標準因子とQ-線形同値なDをとったとき，(X,D)が大域的F分裂である為の十分条件に関するある予想である．

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