Group actions and vector bundles in algebraic geometry

本文研究了代数几何中群作用和向量束的一些问题。该提案分为四个部分。第一部分是研究代数变量上的左向量束。特别地，提议者计划研究他关于这种束的一个猜想，并将Fulton和Lazarsfeld对样本束证明的一些正性结果推广到左束。该提案的第二部分涉及标记有向图，该有向图是在代数变量上为某些环面作用定义的矩图上建模的。定义了这样一个广义矩图的“交同调”庞加莱多项式，目的是看看这个多项式是否对这种类型的更一般的图是回文和单峰的，不一定是作为矩图产生的，但（就可以确定的而言）具有矩图所具有的所有图论性质。本文的第三部分是证明环面等变k理论中关于旗子簇的乘法的一个非负性定理。第四部分是用等变黎曼-罗希映射的代数群来描述几何商的黎曼-罗希映射。代数几何是数学的一个分支，它是从研究多项式方程的解中发展出来的。通过将代数和几何联系起来，它揭示了这两者，并对许多其他数学分支（包括数论、密码学和群论）产生了影响。本课题主要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，这使得它更容易被研究。有时，对称性可以用来提取问题的基本特征，并在不考虑所有几何复杂性的情况下进行研究；这个项目的一部分，力矩图，就是一个例子。

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.