Birational Geometry and Hodge Theory

该项目将分为代数几何的各个子项目。在第一个子项目中，Arapura打算研究消失定理，它大致是控制同调不变量的技术集合。在第二个子项目中，arapuraind倾向于研究与Hodge猜想有关的一些问题，该猜想预测某些称为Hodge类的同调实体可以以几何方式实现。在第五个子项目中，Matsuki计划研究最小模型规划的同调方面，并研究三维空间的自同构。几个子项目涉及到为各种物种寻找好的模型或分辨率，以及它们之间的地图。对于地图，这可以更精确地表述为环形化问题，这将由Matsuki在第四个子项目中研究。为各种各样的奇点找到好的模型就等于解决了奇点的问题。Matsuki和Wlodarczyk将在第七个子项目中研究这个问题的各个方面。特别是，Wlodarczyk发现了一种简化的算法来解决特征零点的奇点，他计划对其进行改进。在第八个项目中，Wlodarczyk将进一步发展他的分层环面品种理论，这将扩展环面嵌入理论。代数变量是数学中的基本对象；它们是代数方程组的解的集合。它们已经在数学物理和密码学等不同领域得到了应用。这个项目的目标是进一步理解这些对象。一种标准的技术涉及到用它们的同调不变量来表示对象，这些不变量通常更容易访问，而且通常是可计算的。一些子项目涉及到这种方法。剩下的子项目包括为代数变量寻找好的模型。

The project will be broken into various subprojects in algebraicgeometry. In the first subproject, Arapura intends to study vanishingtheorems, which are very roughly a collection of techniques forcontrolling homological invariants. In the second subproject, Arapuraintends to study some problems related to the Hodge conjecture, whichpredicts that certain homological entities called Hodge classes can berealized in a geometric way. In the fifth subproject, Matsuki plans tostudy the homological aspects of the minimal model program, and alsoto study the automorphisms of the three dimensional space. Severalsubproject involve finding good models, or resolutions, for varietiesand maps between them. For maps this can be formulated more preciselyas the toroidalization problem, and this will be studied by Matsuki inthe fourth subproject. Finding good models for a variety amounts toresolution of singularities. Various aspects of the problem will bestudied by Matsuki and Wlodarczyk in the seventh subproject. Inparticular, Wlodarczyk has found a simplified algorithm forresolutions of singularities in characteristic zero, which he plans torefine. In the eighth project, Wlodarczyk will further develop histheory of stratified toroidal varieties, which extends the theory oftoroidal embeddings.Algebraic varieties are basic objects in mathematics; they are sets ofsolutions of systems of algebraic equations. They have foundapplications in areas as diverse as mathematical physics andcryptography. The goal of this project is to further the understandingof these objects. A standard technique involves expressing the objectsby their homological invariants, which are usually more accessible andoften computable. Some of the subproject involve this approach. Theremaining subproject involves finding good models for algebraicvarieties.