## What is known about formality of flag varieties?

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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.

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## 1 Answer

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

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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
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