## On Fubini-Study Metrics

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!