Group actions and vector bundles in algebraic geometry

DMS-0403838威廉·A·格雷厄姆这个建议涉及到对代数几何中的一些问题的研究，这些问题是在群作用和向量丛的背景下出现的。该提案包括四个部分。建议的第一部分是研究代数簇上的LEF向量丛。具体地说，提出者计划研究他关于这类丛的一个猜想，并将Fulton和Lazarsfeld证明的一些正性结果推广到LEF丛。该提案的第二部分涉及基于为代数簇上的某些环面作用定义的矩图的标记有向图。定义了这种广义矩图的“交同调”Poincare多项式，目的是确定对于更一般的这种类型的图，这个多项式是否回文和单峰，不一定是矩图，但具有(尽可能确定的)矩图所具有的所有图论性质。第三部分证明了旗族环面等变K-理论中关于乘法的一个非负性定理。建议的第四部分是用等变Riemann-Roch映射来描述几何商的代数群的Riemann-Roch映射。代数几何是从研究多项式方程的解发展而来的数学分支。通过将代数和几何联系起来，它揭示了两者，并对数学的许多其他分支有影响，包括数论、密码学和群论。这个项目主要致力于研究代数几何中与代数群有关的一些问题。说一个问题与代数群有关，意味着这个问题具有某种对称性，这使它更容易被研究。有时对称性可以用来提取问题的基本特征并研究它们，而不必考虑所有的几何复杂性；矩图就是一个例子，它是这个项目的一部分。

DMS-0403838William A. GrahamThis proposal concerns research on some problems in algebraic geometry that arise in the context of group actions and vector bundles. There are four parts to the proposal. The first part of the proposal is to study lef vector bundles on algebraic varieties. In particular the proposer plans to study a conjecture he has made concerning such bundles, and also to extend to lef bundles some positivity results proved by Fulton and Lazarsfeld for ample bundles. The second part of the proposal concerns labeled directed graphs modeled on the moment graphs defined for certain torus actions on algebraic varieties. The``intersection homology'' Poincare polynomial of such a generalized moment graph is defined, and the goal is to see if this polynomial is palindromic and unimodal for more general graphs of this type, not necessarily arising as moment graphs, but with (as far as can be determined) all the graph-theoretic properties that moment graphs have. The third part of the proposal is to prove a non-negativity theorem about the multiplication in the torus-equivariant K-theory of the flag variety. The fourth part of the proposal is to describe the Riemann-Roch map of a geometric quotient by an algebraic group in terms of the equivariant Riemann-Roch map.Algebraic geometry is a branch of mathematics that has grown out of the study of solutions to polynomial equations. By linking algebra and geometry, it sheds light on both, and has implications for many other branches of mathematics, including number theory, cryptography, and group theory. This project is largely devoted to studying some problems in algebraic geometry that are related to algebraic groups. To say that a problem is related to algebraic groups means that the problem has some symmetry which can make it more accessible to investigation. Sometimes the symmetry can be used to extract essential features of the problem and study them without considering all of the geometric complexities; the moment graphs, which are part of this project, are an example of this.