4
$\begingroup$

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.

$\endgroup$

1 Answer 1

4
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ Sullivans proof uses algebraic groups. There is another proof, without algebraic geometry, in the book "Algebraic models in Geometry", by Felix, Oprea, Tanre. $\endgroup$ Apr 30, 2012 at 16:54
  • $\begingroup$ @Ebert Felix, Oprea, Thomas. $\endgroup$ Aug 29, 2019 at 9:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.