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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?

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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
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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
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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
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$h^{11}=1$ for Grasmanians. For flag varieties, it can be bigger. –  Donu Arapura Mar 23 '11 at 17:53
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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
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2 Answers 2

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?

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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 $L$.

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