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Hi,

Can one define a Fubini-Study metric/Kaehler metric on the projective space of an infinite dimensional Hilbert space, i.e. using the formula $\partial \bar{\partial} \log |Z|^2$? This should be very well-known to the experts. Anyhow I don't have much experience with infinite dimension and worried that something may go wrong. I appreciate any comments or references.

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2 Answers 2

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Maybe it's easier to see that the definition extends if you don't use a formula:

The unit sphere inherits a Riemannian metric from the Hilbert space in the standard manner and since it is invariant under the circular symmetry $(e^{i\theta},x) \mapsto e^{i\theta} x$, it will project down to a Riemannian metric on the complex projective space.

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  • $\begingroup$ Thanks Ahmed and Alvarez, so definition works as in the finite dimensional case. $\endgroup$
    – Kiu
    Commented Apr 2, 2013 at 3:20
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The answer is yes and can be found for example in S.Kobayashi "The geometry of bounded domains" T.A.M.S 1959 (92) 267- 290

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  • $\begingroup$ Just a small correction. I assummed taht you mean separable Hilbert space. For wild Hilbert spaces not allowing countable orthonormal basis the above answer does not apply $\endgroup$
    – Ahmed Sulejmani
    Commented Apr 1, 2013 at 13:46

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