陳-韋伊同態

數學上，陳－韋伊同態（英語：Chern–Weil homomorphism）是陳－韋伊理論的基本構造，將一個光滑流形M的曲率聯繫到M的德拉姆上同調群，也就是從幾何到拓撲。這個理論由陳省身和安德烈·韋伊於1940年代建立，是發展示性類理論的重要步驟。這個結果推廣了陳-高斯-博內定理。

記${\displaystyle \mathbb {K} }$為實數域或複數域。設G為實或複李群，有李代數${\displaystyle {\mathfrak {g}}}$，又記

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

為${\displaystyle {\mathfrak {g}}}$上的${\displaystyle \mathbb {K} }$-值多項式的代數。設${\displaystyle \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$為在${\displaystyle \mathbb {K} ({\mathfrak {g}}^{*})}$中G的伴隨作用的不動點的子代數，故對所有${\displaystyle f\in \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$有

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

陳－韋伊同態是從${\displaystyle \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$到上同調代數${\displaystyle H^{*}(M,\mathbb {K} )}$的一個${\displaystyle \mathbb {K} }$-代數同態。這個同態存在，且對M上任何主G-叢P有唯一定義。若G緊緻，則於此同態下，G-叢B^G的分類空間的上同調環同構於不變多項式的代數${\displaystyle \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$：

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

對於如SL(n,R)的非緊緻群，可能有上同調類無不變多項式的表示。

同態的定義

取P 中任何聯絡形式w，設${\displaystyle \Omega }$為相伴的曲率2-形式。若${\displaystyle f\in \mathbb {K} ({\mathfrak {g}}^{*})^{Ad(G)}}$是k次齊次多項式，設 ${\displaystyle f(\Omega )}$是P上的2k-形式，以下式給出

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

其中${\displaystyle \epsilon _{\sigma }}$是2k個數的對稱群${\displaystyle {\mathfrak {S}}_{2k}}$中置換${\displaystyle \sigma }$的符號。（見普法夫值。）

可證

${\displaystyle f(\Omega )}$

是閉形式，故

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

且${\displaystyle f(\Omega )\,}$的德拉姆上同調類獨立於在P上的聯絡的選取，故只依賴於主叢。

因此設

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

是由上從f得出的上同調類，故有代數同態

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

參考

• 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 請檢查|publication-date=中的日期值 (幫助).
• 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.