Welcome to MathOverflow

Question

3

4

Let $G$ be a connected, complex reductive algebraic group and $X=G/P$ a (partial) flag variety. For example by "Real homotopy theory of Kähler manifolds", we know that its cohomology with real coefficients is formal. I have a bunch of questions about other coefficients:

Does real formality automatically imply formality over $\mathbb Q$? Probably not in general, but rational formality of flag varieties should be known right?

Are flag varieties even formal over $\mathbb Z$? I guess not, since the cohomology is more complicated in that case. Are there counterexamples?

What about $\mathbb {CP}^n$? I guess it is formal over $\mathbb Z$, is there a proof written somewhere?

Are flag varieties formal over the integers with say the order of the Weyl group invertible? In this case at least the cohomology ring is isomorphic to the coinvariants.

flag	asked Apr 27 2012 at 21:48 Jan Weidner 5,908●21●39

Johannes Ebert · Answer 1 · 2012-04-27 22:05:31Z UTC

3

Formality over $\mathbb{C}$ implies formality over $\mathbb{Q}$. This is Theorem 12.1. in Sullivan ''Infinitesimal computations in topology''.

answered Apr 27 2012 at 22:05

Johannes Ebert
11.5k●33●55

	Thanks, this is good to know! – Jan Weidner Apr 30 2012 at 15:18
	Sullivans proof uses algebraic groups. There is another proof, without algebraic geometry, in the book "Algebraic models in Geometry", by Felix, Oprea, Tanre. – Johannes Ebert Apr 30 2012 at 16:54

What is known about formality of flag varieties?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.