Linear systems on fibrations

纤维线性系统

• 批准号: 1400943
• 负责人: Vyacheslav Shokurov
• 金额: $ 18万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1400943&HistoricalAwards=false
• 关键词: Linear; systems; fibrations

This award supports a research project in the field of algebra and geometry, old and traditional areas of mathematics, with methods and applications in modern birational geometry. These methods and applications interact with most of branches of mathematics, including differential geometry, topology, number theory, and algebra, and can be useful in mathematical physics, cosmology, cryptography, and robotics. From its origin geometry is closely related to volume. The main objects under investigation are differential forms of volume type on families and their products. The project addresses problems on polynomial finite generatedness of those forms and on estimation of degrees of generators with respect to the dimension and possibly to some other discrete invariants of members of families. New techniques are needed in dealing with differential forms on families or relative differential forms. Another novelty of the project is to develop theory of the finite generatedness based on universal mappings from moduli theory instead of traditionally used vanishing of cohomologies.The project deals with fundamental problems on linear systems of divisors associated to twisted differential forms on fibered spaces. Standard problems about linear systems are problems about their base points, about singularities of their general or special members, and about correspondences with or isomorphisms to other systems. Respectively, the PI considers the semiampleness problem for moduli part of weakly log canonical families, the construction of log canonical complements for fibered varieties with applications to log canonical local and global thresholds, to a new beta invariant and to Tian's alpha invariant, and explores possibility of an isomorphism between log canonical linear systems adjoint to the discriminant curve for different conic bundles structures of a threefold with applications to birational classification of those conic bundles. Strictly fibered cases for log canonical complements and for Sarkisov's links do not cover all possible cases, they cover main cases and remaining ones are exceptional. The latter form usually bounded families up to birational isomorphisms. A new technique relevant to these problems will be developed under the project. This makes use of relative toroidal log singularities, theory of b-polarizations, theory of moduli of triples, interlaced by flops, and correspondences of those moduli. Investigation of moduli of varieties defined up to flops is natural and agrees with the spirit of current birational geometry where the minimal models are defined up to flops.

该奖项支持代数和几何领域的研究项目，数学的古老和传统领域，以及现代双有理几何的方法和应用。这些方法和应用与大多数数学分支相互作用，包括微分几何，拓扑学，数论和代数，并且可以在数学物理，宇宙学，密码学和机器人学中有用。从它的起源几何是密切相关的体积。研究的主要对象是族及其乘积的体积类型的微分形式。该项目解决的问题多项式有限generatedness的这些形式和估计度的发电机方面的尺寸和可能的一些其他离散不变量的家庭成员。在处理族上的微分形式或相对微分形式时，需要新的技巧。该项目的另一个新奇是发展理论的有限generatedness的基础上普遍映射的模理论，而不是传统上使用消失的cohomologies.The项目涉及的基本问题，线性系统的因子相关的扭曲微分形式的纤维空间。关于线性系统的标准问题是关于它们的基点、关于它们的一般或特殊成员的奇异性、关于与其他系统的对应或同构的问题。分别地，PI考虑了弱对数典型族的模部分的半满问题，纤维簇的对数典型补的构造及其对对数典型局部和全局阈值的应用，对新的beta不变量和Tian的alpha不变量，并探讨了三重锥束不同结构的判别曲线的对数正则线性系统之间同构的可能性。并应用于锥丛的双有理分类。严格纤维的情况下，日志典型的补充和Sarkisov的链接不包括所有可能的情况下，他们涵盖了主要的情况下，其余的是例外。后者的形式通常有界的家庭双理性同构。将在该项目下开发与这些问题有关的新技术。这使得使用相对环形日志奇异性，理论的b-极化，理论的三重模，交错的触发器，和对应的这些模量。调查的模量定义的品种，以触发器是自然的，并同意目前的双有理几何的精神，其中最小的模型被定义为触发器。