Birational Geometry and Hodge Theory

该项目将被分解成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，这是非常粗略的控制同调不变量的技术的集合。在第二个子项目中，Arapura试图研究与Hodge猜想有关的一些问题，Hodge猜想预言某些称为Hodge类的同调实体可以用几何的方式实现。在第五个子项目中，Matsuki计划研究最小模型程序的同调方面，并研究三维空间的自同构。几个子项目涉及到为品种和它们之间的图谱找到好的模型或分辨率。对于地图，这可以制定更精确的toroidalization问题，这将是研究Matsuki在第四子项目。找到好的模型就等于解决了奇异性问题。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发现了一个简化的算法为决议的奇异性在特征零，他计划完善。在第八个项目中，Wlodarczyk将进一步发展他的分层环形簇理论，它扩展了环形嵌入理论。代数簇是数学中的基本对象，它们是代数方程组的解集。 它们在数学、物理和密码学等不同领域都有应用。这个项目的目标是进一步了解这些对象。一个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理解，也更容易计算。一些子项目涉及这种方法。 剩下的子项目涉及到为代数变种寻找好的模型。

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.