Fubini-Study Metric

The Fubini-Study metric is the standard Kähler metric on complex projective space, induced from the Hermitian inner product on its homogeneous coordinates. In the affine chart , one normalization has Kähler potential and Hermitian coefficient matrix

The associated Kähler form is . Constant rescaling changes the curvature normalization, so numerical curvature statements must specify the scale. Restricting this form to a smooth complex projective submanifold gives its standard induced symplectic form.