Birational Geometry and Hodge Theory

这个项目分为几个部分，涉及代数几何的几个相互关联的领域，以两族几何和代数变异的霍奇理论为中心。在第一部分中，研究者打算在代数几何中出现的某些普遍空间上构造微分，并将其应用于代数循环的研究。在第二部分中，研究者与D. abramovich和K. Karu合作，打算扩展他们之前的工作，以证明两国地图的强因子分解猜想。这一猜想认为，光滑完全变种之间的任何两族图都有一个特别简单的结构：它是一个连续的爆炸，接着是一个以光滑为中心的一系列爆炸。在第三部分中，研究者将应用先前建立的弱分解猜想来比较两个双等效最小模型的Hodge结构。在第四部分中，一位研究者打算推广他们之前的消失定理，并将其应用于双国不变量的研究。在第五部分，一位研究者将尝试把高同伦群的Hodge理论与代数环的交理论联系起来。在第六部分中，一位研究者打算在正特征域上研究一类曲面奇点，这类曲面奇点在双国几何中很重要。在第六部分和最后一部分中，一位研究者打算通过考虑某些分层来扩展环形嵌入理论。代数变量是一种几何对象，它为数学以及物理学和计算机科学等邻近科学领域的许多现象提供了丰富的模型。它们的优点是可以用有限项来描述，就像有限代数方程组的解一样。然而，这些描述往往是复杂的，并不是唯一的；判断两个这样的描述何时会导致相等，甚至近似相等的变体是非常困难的。近似等价的概念使近似等价变得精确。本课题的目标之一是研究双元等价关系的精细结构。本课题的另一个目标是引入和研究某些自然双族不变量，即代数变量几何复杂性的度量。其中一些不变量计算与代数变化相关的调和（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，两族映射的精细结构将产生对这些不变量性质的见解。

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.