For the projective line $CP^1$, its cohomology ring has a single generator. Moreover, this generator is given by the cohomology class of the fundamental form associated associated to the Fubini--Study Kahler metric on $CP^1$. If I have understood correctly this also holds for all the higher projective spaces.


Now the well-known Borel theorem for the more flag varieties says that in general the cohomology ring of a flag variety has multiple generators. Do these generators come from Kahler metrics too? Or is there any sense in which the $CP^n$ case generalises to a more general result?



Kahler Metric Fundamental Forms and Cohomology Ring Generators
Antonio,


The answer to first question is no. Let $F$ be the variety of full flags in $\mathbb{C}^3$. This can be viewed as variety of
pairs $F=\lbrace (p,\ell)\in \mathbb{C}\mathbb{P}^2\times \mathbb{C}\check{\mathbb{P}}^2\mid p\in \ell\rbrace$. There are two generators for cohomology given by pulling back the Kaehler class from $\mathbb{C}\mathbb{P}^2$, or its dual, under both projections. Neither of these are Kaehler forms on $F$, because they aren't positive along the fibres of the projection. What is special about projective space is that the second Betti number is $1$, so $H^2$ has to be generated by a Kaehler class.


