Fubini-Study metric

The Fubini-Study metric is the standard unitary-invariant Kähler metric on Complex projective space, induced from the unit sphere by the Hopf fibration. Its associated Kähler form is the Fubini-Study form; scaling conventions determine the area of a projective line.

The Fubini-Study form is the standard Kähler form on complex projective space. With the integral normalization, $[{\omega }_{FS}/(2\pi )]=c_1(\mathcal{O}(1))$.

The two-form is associated with the Fubini-Study metric.

The form ${\omega }_{\mathrm{int}}=\frac{i}{2\pi }\partial \bar{\partial }\log (1+\sum |z_j{|}^2)$ patches on Complex projective space. The dual tautological metric on $\mathcal{O}(1)$ identifies it with normalized Chern curvature, so $[{\omega }_{\mathrm{int}}]=c_1(\mathcal{O}(1))$. With the unnormalized convention $i\partial \bar{\partial }\log (1+\sum |z_j{|}^2)$, divide by $2\pi$ to obtain this integral class.

The scalar circle group action on ${\mathbb{C}}^{n+1}$ with the standard symplectic form has moment map $\mu =-|z{|}^2/2$ in the convention ${\iota }_{X_H}\omega =dH$. Reduction of the radius-$R$ sphere gives Complex projective space and the form ${\omega }_R=(i{R}^2/2)\partial \bar{\partial }\log (1+|w{|}^2)$. Its area on a complex projective line is $\pi R^2$. Thus $R=\sqrt{2}$ yields the Fubini-Study form normalized to projective-line area $2\pi$, while $R=1$ yields half that form. The Hopf fibration identifies the quotient.

Normalize the Fubini-Study form so that a complex projective line has symplectic area $\pi$. The displayed map takes the unit symplectic ball in ${\mathbb{C}}^n$ diffeomorphically onto the complement of the last-coordinate hyperplane. Its lift to the unit sphere has last coordinate positive real, so that coordinate contributes zero to the pulled-back standard symplectic form. The map is therefore a symplectomorphism.

Two disjoint symplectic balls of radii $r_1,r_2$ embedding into the unit four-ball must satisfy the displayed inequality. Extend their transported standard complex structures on smaller concentric balls to a compatible almost complex structure on the projective plane. A degree-one J-holomorphic curve through both centres has area $\pi$. The Monotonicity theorem for a J-holomorphic curve bounds its area in each smaller ball below by $\pi {\rho }_i^2$. Disjointness allows these bounds to be added; letting ${\rho }_i$ increase to $r_i$ proves the result.