Equivariant Cohomology in Algebraic Geometry Lecture Two: Definitions and Basic Properties William Fulton Notes by Dave Anderson
نویسنده
چکیده
For a Lie group G, we are looking for a right principal G-bundle EG → BG, with EG contractible. Such a bundle is universal in the topological setting: if E → B is any principal G-bundle, then there is a map B → BG, unique up to homotopy, such that E is isomorphic to the pullback of EG. See [Hus75] for the existence of these universal principal bundles; we will not need the general story here. We will also find principal G-bundles EGm → BGm, with πi(EGm) = 0 (and H (EGm) = 0) for 0 < i < k(m), where k(m) goes to infinity as m grows. For such bundles, we have H i GX := H (EG× X) = H (EGm × G X) for i < k(m). To see this, we need the following proposition: Proposition 1.1. If E → B and E → B are two principal right Gbundles, and H (E) = H (E) = 0 for 0 < i < k, then there is a canonical isomorphism H (E × X) ∼= H (E × X) for i < k. Proof. Let G act diagonally on E × E, so there is a diagram E ×X E × E ×X E ×X
منابع مشابه
Introduction to Equivariant Cohomology in Algebraic Geometry
Introduced by Borel in the late 1950’s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. Quite some time passed before algebraic geometers picked up on these ideas, but in the last twenty years, equivariant techniques have found many applications in enumerative geometry, Gromov-Witten theory, and the study of toric varieties and homogen...
متن کاملEquivariant Cohomology in Algebraic Geometry Lecture Five: Projective Space; Localization Ii
متن کامل
Equivariant Cohomology in Algebraic Geometry Lecture Nine: Flag Varieties
Remark 0.1. We saw that the study of cohomology and equivariant cohomology of Grassmannians leads to interesting symmetric polynomials, namely, the Schur polynomials sλ(x) and sλ(x|t). These arise in contexts other than intersection theory and representation theory. For example, Griffiths asked which polynomials P in c1(E), . . . , cn(E) are positive whenever E is an ample vector bundle on an n...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007