MathOverflow Asked by ThiKu on January 5, 2021
For a hermitian symmetric space $M$ one has its group of biholomorphic maps $operatorname{Hol}(M)$ and its group of Riemannian isometries $operatorname{Isom}(M)$. According to Prop. 1.6 of Milne – Introduction to Shimura varieties, the inclusion of their intersection into either of them yields an isomorphism of identity components.
Looking at $S^2=operatorname P^1mathbb C$ this seems to mean that $operatorname{PSL}(2,mathbb C)$ is isomorphic to $operatorname{SO}(3)$, which is of course wrong. So what am I misunderstanding here?
To get this question off the unanswered list:
In Proposition 1.6 Milne assumes that $X$ is a Hermitian symmetric domain, equivalently, a Hermitian symmetric space of noncompact type. This assumption rules out examples such as complex-projective spaces and complex-affine spaces as well as spaces containing such direct factors (for all of these the conclusion of the proposition fails).
Correct answer by Moishe Kohan on January 5, 2021
Get help from others!
Recent Questions
Recent Answers
© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP