Group actions and vector bundles in algebraic geometry

DMS-0403838 William A.格雷厄姆这一建议涉及研究的一些问题，代数几何中出现的背景下，组行动和向量束。 该建议共分四个部分。 第一部分是研究代数簇上的左向量丛。 特别是提议者计划研究一个猜想，他提出了关于这种束，并扩大到左束的一些积极性的结果证明了富尔顿和Lazarsfeld充分束。该提案的第二部分涉及标记有向图建模的时刻图定义为某些环面行动代数簇。 定义了这样一个广义矩图的"相交同源“庞加莱多项式，目标是看看这个多项式对于这种类型的更一般的图是否是回文的和单峰的，不一定是矩图，但具有（尽可能确定）矩图所具有的所有图论性质。第三部分证明了旗簇环面等变K-理论中乘法的一个非负性定理。建议的第四部分是用等变Riemann-Roch映射来描述代数群的几何商的Riemann-Roch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发展而来的。 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，这可以使它更容易被研究。 有时，对称性可以用来提取问题的基本特征，并在不考虑所有几何复杂性的情况下研究它们;作为该项目一部分的矩图就是一个例子。

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.