Fubini–Study metric - Imported Knowlpedia Content

Definition

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 $U(n+1)$ ; this construction is presented in Demailly, Chapter VI, §4, Example 4.4.

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. Griffiths and Harris treat it as the canonical projective metric in Chapter 0, §5.

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.