On Fubini-Study Metrics

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1. A Little Complex Analysis

We want to introduce the notion of a ‘Fubini-Study’ metric which is important in Complex Manifold Theory and Differential Geometry (and the associated theories such as Mathematical Physics). But first we need to introduce a little Complex Analysis. The source is of course Griffiths and Harris. Let M be a complex manifold, {p \in M} any point, and {z=(z_{1},\cdots,z_{n})} a holomorophic co-ordinate system around p. There are three different notions of a tangent space to M at p,which we now describe:

  • {T_{\mathbb{R},p}(M)} is the usual real tangent space to M at p,when we consider M a real manifold of dimension 2n. {T_{\mathbb{R},p}(M)} can be realized as the space of {\mathbb{R}-}linear derivations on the ring of real-valued {C^{\infty}}-functions in a neighbourhood of p; if we write {z_i = x_i + iy_i}, {T_{\mathbb{R},p}(M) = \mathbb{R}(\frac{\partial}{\partial x_{i}}, \frac{\partial}{\partial y_i}}.
  • {T_{\mathbb{C},p}(M) = T_{\mathbb{R},p}(M)\otimes_{\mathbb{R}} \mathbb{C}} is called the complexified tangent space to M at p. It can be realized as the space of {\mathbb{C} -} linear derivations in the ring of complex valued {C^{\infty}}-functions on M around p. We can write {T_{\mathbb{C},p}(M) = \mathbb{C}{\frac{\partial}{\partial x_{i}},\frac{\partial}{\partial y_i}}}
    ={\mathbb{C}{\frac{\partial}{\partial z_{i}},\frac{\partial}{\partial \bar{z}_i}}}

  • {T'_p(M)= \mathbb{C}{\frac{\partial}{\partial z_{i}}}\subset T_{\mathbb{C}, p}(M)} is called the holomorphic tangent space to M at p. It can be realized as the subspace of {T_{\mathbb{C},p}(M)} consisting of derivations that vanish on antiholomorphic functions (i.e. F such that T is holomorphic), and so is independent of the holomorphic co-ordinate system chosen. The subspace {T''_p(M)= \mathbb{C}{\frac{\partial}{\partial \bar{z}_{i}}}} is called the antiholomorphic tangent space to M at p; clearly {T_{\mathbb{C},p}(M) = T'_p(M) \oplus T''_p(M)}

Now we consider some Calculus on Complex Manifolds. Let M be a complex manifold of dimension n. A hermitian metric on M is given by a positive definite hermitian inner product {(,)_z: T'_z(M) \otimes T'_z(M) \rightarrow \mathbb{C}} on the holomorphic tangent space at z for each {z \in M},
depending smootly on z – that is, such that for local co-ordinates z on M the function
{h_ij(z) = (\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_j})_z} are {C^{\infty}}

Writing {(,)_z} in terms of the basis {{dz_i \otimes d\bar{z}_j}} for {(T'_z(M) \otimes \bar{T'_z(M)}^{\textasteriskcentered} = T^{\textasteriskcentered\textquoteright}_z(M) \otimes T^{* \textquotedblright}_{z}(M)}, the hermitian metric is given by {ds^{2} = \sum_{i,j} h_{ij}(z) dz_i \otimes d \bar{z}_j} So let us describe the Fubini-Study Metric Let {z_0,\cdots,z_n} be co-ordinates on {\mathbb{C}^{n+1}} and denote by {\pi:\mathbb{C}^{n+1} -{0} \rightarrow \mathbb{P}^n} the standard projection map. Let {U \subset \mathbb{P}^{n}} be an open set and {Z: U \rightarrow \mathbb{C}^{n-1} - {0}} a lifting of U, i.e. a holomorphic map with {\pi \circ z = id}; consider the differential form
{\omega = \dfrac{i}{2\pi}\partial \bar{\partial}log\|z\|^{2}} If {Z':U \rightarrow \mathbb{C}^{n-1} - {0}} is another lifting, then {Z' = f.Z} with f a nonzero holomorphic function, so that
{\dfrac{i}{2\pi}\partial \bar{\partial}log\|z\|^{2} = \frac{i}{2 \pi}\partial \bar{\partial} (log\|z\|^{2} + log f + log \tilde{f})}
{= \omega + \dfrac{i}{2\pi}(\partial \bar{\partial}log f - \bar{\partial} \partial log \tilde{f})} = {\omega} Therefore {\omega}is independent of the lifting chosen; since liftings always exist locally, {\omega} is a globally defined differential form in {\mathbb{P}^{n}}. (By the sheaf properties of differential forms) Clearly {\omega} is of type (1,1). To see that {\omega} is positive, first note that the unitary group {U(n+1)} acts transitively on {\mathbb{P}^{n}} and leaves the form {\omega} positive everywhere if it is positive at one point. Now let {{w_i = z_i/z_0}} be co-oridnates on the open set {U_{0} = (z_0 \neq 0)}in {\mathbb{P}^{n}} and use the lifting {Z = (1,w_1,\cdots,w_n)} on {U_0} ; we have (after some substitutions

{\omega = \dfrac{i}{2 \pi} [\frac{\sum dw_i \wedge d\bar{w}_i}{1 + \sum w_i \bar{w}_i} - \frac{(\sum \bar{w}_i dw_i \wedge \sum w_i d\bar{w}_i)}{(1 + \sum w_i \bar{w}_i)^{2}}]} At the point {[1,0,\cdots,0]}, \\ {\omega = \frac{i}{2\pi} \sum dw_i \wedge d \bar{w}_i > 0} Thus {\omega} defines a particular hermitian metric on the projective complex space called the Fubini-Study metric. That was the aim of the article!