{"title":"Positivity of Schur forms for strongly decomposably positive vector bundles","authors":"Xueyuan Wan","doi":"10.1017/fms.2023.125","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum of Mathematics Sigma","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/fms.2023.125","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.
在本文中，我们定义了两类强可分解正定性，它们是（对偶）中野正定性的概括，比 S. 芬斯基引入的可分解正定性更强。我们提供了 I 型和 II 型强可分解正性的标准，并证明了 I 型强可分解正向量束的舒尔形式是弱正性的，而 II 型强可分解正向量束的舒尔形式是正性的。这肯定地回答了格里菲斯关于强可分解正向量束的一个问题。因此，我们对（对偶）中野正向量束的舒尔形式的实在性提出了代数证明，这最初是由 S. 芬斯基证明的。
期刊介绍：
Forum of Mathematics, Sigma is the open access alternative to the leading specialist mathematics journals. Editorial decisions are made by dedicated clusters of editors concentrated in the following areas: foundations of mathematics, discrete mathematics, algebra, number theory, algebraic and complex geometry, differential geometry and geometric analysis, topology, analysis, probability, differential equations, computational mathematics, applied analysis, mathematical physics, and theoretical computer science. This classification exists to aid the peer review process. Contributions which do not neatly fit within these categories are still welcome.
Forum of Mathematics, Pi and Forum of Mathematics, Sigma are an exciting new development in journal publishing. Together they offer fully open access publication combined with peer-review standards set by an international editorial board of the highest calibre, and all backed by Cambridge University Press and our commitment to quality. Strong research papers from all parts of pure mathematics and related areas will be welcomed. All published papers will be free online to readers in perpetuity.
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