Birational Geometry and Hodge Theory

这个项目分为几个部分，涉及以双曲几何和代数簇的霍奇理论为中心的几个相互关联的代数几何领域。在第一部分中，研究人员打算在代数几何中产生的某些泛空间上构造微分，并将这些微分应用于代数圈的研究。在第二部分中，研究人员与D.Abramovichan K.Karu合作，打算扩展他们先前的工作来证明二元映射的强因式分解猜想。这一猜想表明，光滑完备簇之间的任何二元映射都有一个特别简单的结构：它是一个序列爆破，然后是一系列具有光滑中心的打击。在第三部分中，研究者将应用先前建立的弱因式分解猜想来比较两个出生等价极小模型的Hodge结构。在第四部分中，一位研究者打算推广他们以前的消零定理，并将它们应用到出生不变量的研究中。在第五部分，其中一位研究者将尝试将高次同伦群的Hodge理论与代数圈的交集理论联系起来。在第六部分中，一位研究者打算研究正特征域上的一类曲面奇点，这类奇点对于二次几何是重要的。在第六部分，也就是最后一部分，其中一位研究人员打算通过考虑某些分层来扩展环形嵌入理论。代数簇是几何对象，它为数学以及物理和计算机科学等邻近科学领域中的许多现象提供了一套丰富的模型。它们的优点是可以用有限项描述，作为有限代数方程组的解。然而，这些描述往往是复杂的，而且不是唯一的；确定两个这样的描述何时会导致等价或甚至近似等价的变体是非常困难的。本课题的目的之一是研究代数簇几何等价关系的更精细结构，另一个目标是引入和研究某些自然的代数簇不变量，即代数簇的几何复杂性的度量。其中一些不变量计算与代数变量相关的调和(能量最小化)对象的数量。这两个目标是相关的，因为研究人员期望两个映射的精细结构将产生对这些不变量的性质的洞察。

This project, which is divided into several parts,is concerned with several interrelated areas ofalgebraic geometry centered around the birational geometry and the Hodge theory of algebraic varieties. In the first part,the investigators intend to construct differentials on certain universal spaces arising in algebraic geometry, and apply these tothe study of algebraic cycles. In the second partthe investigators, in collaboration with D. Abramovichand K. Karu, intend to extend their previous work to theprove the strong factorization conjecture for birational maps.This conjecture says that any birational map between smoothcomplete varieties has a particularly simple structure: itis a sequence blow ups followed by a sequence of blow downswith smooth centers. In the third part, the investigatorswill apply the previously established weak factorization conjecture to compare the Hodge structure of two birationally equivalent minimal models. In the fourth part, one of the investigators intends to extend their previous vanishingtheorems and apply them to the study of birationalinvariants. In the fifth part, one of the investigators will attempt to relate the Hodge theory of higher homotopy groups to the intersection theory of algebraic cycles. In thesixth part, one of the investigators intends to study a class of surface singularities, which are important for birationalgeometry, over fields of positive characteristic. In thesixth and final part, one of the investigators intends to extend the theory of toroidal embeddings by taking into accountcertain stratifications. Algebraic varieties are geometric objects which provide arich set of models for a number of phenomena within mathematicsas well as in neighboring fields of science such as physics andcomputer science. They have the advantage of being describable in finite terms, as solutions to a finite system of algebraicequations. However, these descriptions are often complicated and not unique; deciding when two such descriptions lead to equivalent,or even approximately equivalent, varieties is very difficult.Approximate equivalence is made precise by the notion ofbirational equivalence. One of the goals of this project is to study the finer structure of the birational equivalence relation.Another goal of this project is to introduce and study certainnatural birational invariants, that is, measuresof the geometric complexity of algebraic varieties. Some ofthese invariants count the number of harmonic (energyminimizing) objects associated to the algebraic variety.These two goals are related since the investigators expect that thefine structure of the birational maps will yield insights intothe properties of these invariants.