Fubini–Study metric

On complex projective space $\mathbb{CP}^n$ , the Fubini–Study metric is the Kähler metric whose Kähler form on the affine chart $Z_0\neq0$ , with $z_j=Z_j/Z_0$ , is

\omega_{\mathrm{FS}}=i\,\partial\bar\partial\log(1+\lVert z\rVert^2).

Equivalently, its Hermitian coefficients are

(g_{j\bar k})= \frac{(1+\lVert z\rVert^2)\delta_{jk}-\bar z_jz_k} {(1+\lVert z\rVert^2)^2}.

These local formulas agree on chart overlaps and define a smooth positive form. This normalization is fixed in the core; authors also multiply the form by $1/2$ , $1/2\pi$ , or another positive constant.

Global construction and invariance

The local potential $\log(1+\lVert z\rVert^2)$ is the chart expression of $\log\lVert Z\rVert^2$ in homogeneous coordinates. Changing a homogeneous representative adds the logarithm of the squared modulus of a nowhere-zero holomorphic function, whose $i\partial\bar\partial$ vanishes. Hence the forms glue globally. The resulting metric is invariant under the projective action of $U(n+1)$ .

Geometry and normalization

The Fubini–Study metric has positive constant holomorphic sectional curvature, with the numerical value depending on normalization. Its Kähler class generates $H^2(\mathbb{CP}^n;\mathbb Z)$ after the standard integral normalization $[\omega_{\mathrm{FS}}/(2\pi)]$ . On $\mathbb{CP}^1$ , it is a constant multiple of the round metric under the identification with the two-sphere.

The metric is also obtained by Kähler reduction of the unit sphere in $\mathbb C^{n+1}$ by the scalar $S^1$ -action.

References

Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, Wiley, 1978. Wiley DOI record. Relevant: Chapter 0, §5.
Jean-Pierre Demailly, Complex Analytic and Differential Geometry, 2012. Author-hosted text. Relevant: Chapter VI, §4, Example 4.4.