陈-韦伊同态

数学上，陈－韦伊同态（英语：Chern–Weil homomorphism）是陈－韦伊理论的基本构造，将一个光滑流形M的曲率联系到M的德拉姆上同调群，也就是从几何到拓扑。这个理论由陈省身和安德烈·韦伊于1940年代建立，是发展示性类理论的重要步骤。这个结果推广了陈-高斯-博内定理。

记${\displaystyle \mathbb {K} }$为实数域或复数域。设G为实或复李群，有李代数${\displaystyle {\mathfrak {g}}}$，又记

${\displaystyle \mathbb {K} ({\mathfrak {g}}^{*})}$

为${\displaystyle {\mathfrak {g}}}$上的${\displaystyle \mathbb {K} }$-值多项式的代数。设${\displaystyle \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$为在${\displaystyle \mathbb {K} ({\mathfrak {g}}^{*})}$中G的伴随作用的不动点的子代数，故对所有${\displaystyle f\in \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$有

${\displaystyle f(t_{1},\dots ,t_{k})=f(Ad_{g}t_{1},\dots ,Ad_{g}t_{k})\,}$。

陈－韦伊同态是从${\displaystyle \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$到上同调代数${\displaystyle H^{*}(M,\mathbb {K} )}$的一个${\displaystyle \mathbb {K} }$-代数同态。这个同态存在，且对M上任何主G-丛P有唯一定义。若G紧致，则于此同态下，G-丛B^G的分类空间的上同调环同构于不变多项式的代数${\displaystyle \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$：

${\displaystyle H^{*}(B^{G},\mathbb {K} )\cong \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}.}$

对于如SL(n,R)的非紧致群，可能有上同调类无不变多项式的表示。

同态的定义

取P 中任何联络形式w，设${\displaystyle \Omega }$为相伴的曲率2-形式。若${\displaystyle f\in \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$是k次齐次多项式，设 ${\displaystyle f(\Omega )}$是P上的2k-形式，以下式给出

${\displaystyle f(\Omega )(X_{1},\dots ,X_{2k})={\frac {1}{(2k)!}}\sum _{\sigma \in {\mathfrak {S}}_{2k}}\epsilon _{\sigma }f(\Omega (X_{\sigma (1)},X_{\sigma (2)}),\dots ,\Omega (X_{\sigma (2k-1)},X_{\sigma (2k)}))}$

其中${\displaystyle \epsilon _{\sigma }}$是2k个数的对称群${\displaystyle {\mathfrak {S}}_{2k}}$中置换${\displaystyle \sigma }$的符号。（见普法夫值。）

可证

${\displaystyle f(\Omega )}$

是闭形式，故

${\displaystyle df(\Omega )=0,\,}$

且${\displaystyle f(\Omega )\,}$的德拉姆上同调类独立于在P上的联络的选取，故只依赖于主丛。

因此设

${\displaystyle \phi (f)\,}$

是由上从f得出的上同调类，故有代数同态

${\displaystyle \phi :\mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}\rightarrow H^{*}(M,\mathbb {K} ).\,}$

参考

• Bott, R., On the Chern–Weil homomorphism and the continuous cohomology of Lie groups, Advances in Math, 1973, 11: 289–303, doi:10.1016/0001-8708(73)90012-1.
• Chern, S.-S., Topics in Differential Geometry, Institute for Advanced Study, mimeographed lecture notes, 1951.
• Shiing-Shen Chern, Complex Manifolds Without Potential Theory (Springer-Verlag Press, 1995) ISBN 0-387-90422-0, ISBN 3-540-90422-0.

  The appendix of this book: "Geometry of Characteristic Classes" is a very neat and profound introduction to the development of the ideas of characteristic classes.

• Chern, S.-S.; Simons, J, Characteristic forms and geometric invariants, The Annals of Mathematics. Second Series, 1974, 99 (1): 48–69, JSTOR 1971013.
• Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry, Vol. 2, Wiley-Interscience, 1963new ed. 2004 请检查|publication-date=中的日期值 (帮助).
• Narasimhan, M.; Ramanan, S., Existence of universal connections, Amer. J. Math., 1961, 83: 563–572, JSTOR 2372896, doi:10.2307/2372896.
• Morita, Shigeyuki, Geometry of Differential Forms, A.M.S monograph, 2000, 201.