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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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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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