Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

For projective $N$-space $CP^{N}$, there is a canonical Kähler metric called the Fubini-Study metric. Do there exist other Kähler metrics for $CP^N$. If so, is there any classification of such metrics?

More generally, how does this work for the Grassmanians, or even flag manifolds?

share|improve this question
Any metric on $\mathbb{P}^1$ is Kahler. There are uncountably many. The Fubini-Study is homogenous, which is a distinguishing feature. –  Donu Arapura Mar 23 '11 at 16:04
Whenever you have a Kahler metric $\omega$ on a complex manifold, then for $f$ any plurisubharmonic function, the form $\tilde \omega = \omega + i \partial \bar \partial f$ will be another Kahler metric. –  Spiro Karigiannis Mar 23 '11 at 16:51
The Kahler cone of the projective space is one-dimensional, so any Kahler metric $\omega$ on $\mathbb P^n$ will be of the form $\omega = c \omega_{FS} + i \partial \bar \partial \phi$. Here $\omega_{FS}$ is the Fubini-Study metric, $c$ is a positive real number and $\phi$ is a (constant or slightly negative-definite) quasi-plurisubharmonic function. I'm not sure if the same is true for Grassmannians. It depends on their $h^{1,1}$ Hodge number; if it's 1, then it works out the same way, if not, something interesting might happen. –  Gunnar Magnusson Mar 23 '11 at 17:47
$h^{11}=1$ for Grasmanians. For flag varieties, it can be bigger. –  Donu Arapura Mar 23 '11 at 17:53
More generally, $H^2(G,\mathbb Z)=\mathbb Z$ for every complex Grassmannian $G$. The (ample) generator is given by the first Chern class of the determinant of tautological quotient bundle. –  diverietti Mar 23 '11 at 21:40
add comment

2 Answers

up vote 11 down vote accepted

Every complex manifold that admits one Kahler metric $w$ admits a lot of them, indeed $w+i\partial \bar \partial f$ is a Kahler metric if the second derivatives of $f$ are not too large. This is why, asking if one can classify Kahler metrics on $\mathbb CP^n$ is more-less equivalent to ask if one can classify functions on $\mathbb CP^n$. Can we classify functions? It depends on what you want to know.

Even if we want to classify Kahler metrics on $\mathbb C^n$, what can this mean? One analogy can be helpful here. Namely this question is somewhat similar to asking if we can classify convex functions on $\mathbb R^n$. Such a function $f$ always define a Hessian metric on $\mathbb R^n$ given by $g_{ij}=\frac{\partial^2 f}{\partial x_i \partial x_j}$. So, can we classify convex functions?

share|improve this answer
add comment

This is not a classification, but you can get a grip on the space of Kahler metrics on $CP^N$ using Bergman metrics and the Segre embeddings.

To explain this conside let $\{s_\alpha\}$ be a basis of homogeneous polynomials of degree k in N+1 variables. This gives an embedding $CP^N\to CP(H^0(O(k)))$. Now define a metric on H^0(O(k)) by declaring that the s_i are orthonormal. This defines a Fubini-Study metric on P(H^0(O(k)) and which pulls back to a metric on $\omega'$ on $CP^N$.

Now it can be proved that any Kahler metric $\omega$ on $CP^N$ is the limit of metrics of the form $k^{-1}\omega'$ for suitable bases $\{s_\alpha\}$ as $k$ tends to infinity (You can take the $\{s_\alpha\}$ to be orthonormal with respect to the $L^2$-metric induced by $\omega$).

In fact there is nothing special about $CP^N$ here. The above works for any projective manifold $X$ with ample line bundle $L$.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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