Algebraic Varieties, Birational Geometry and the Structure of the Galois Groups

该项目致力于代数簇的二元不变量的泛空间理论。其中一个基本问题是用交换群的射影作用来刻画射影空间乘积的一类自然商集，其中交换群具有定义在有限域的代数闭包上的簇的所有有限双调不变量。PI要解决的第二个更困难的问题是找到一类类似的泛簇，它既适用于有限的泛簇，又适用于定义在数域上的无限代数簇的L不变量。Galois泛函理论提供了一系列结果，可以通过群上同调来描述代数簇的上同调双调不变量。因此，这个项目解决了代数几何、伽罗瓦理论和群论的交界处的问题。PI将把非分枝Brauer群的高维类似的函数场的有限双不变量与有限交换群中交换子群的特殊系统的不变量联系起来。

The project addresses the theory of universal spaces for birational invariants of algebraic varieties. One of the fundamental problems concerns the description of a natural class of quotients of products of projective spaces by the projective actions of abelian groups which held all finite birational invariants of varieties defined over algebraic closures of finite fields. The second more difficult problem which PI is going to address is to find a similar class of universal varieties for finite and then for infinite l-adic invariants of algebraic varieties defined over number fields.The Galois theory of functional fields provides with a series of results allowing to describe the cohomological birational invariants of algebraic varieties through the group cohomology. Thus this project addresses the problems at the interface of algebraic geometry, Galois theory and group theory.The PI is going to relate finite birational invariants of functional fields which are higher dimensional analogues of the nonramified Brauer group to the invariants of special systems of abelian subgroups in finite abelian groups.