nLab Fubini-Study metric

Idea

There is a unique (up to a scalar) hermitian metric on complex projective space (which may be normalized), the Fubini-Study metric.

All analytic subvarieties of a complex projective space are in fact algebraic subvarieties and they inherit the Kähler manifold structure from the projective space.

Examples include complex tori $\mathbb{C}^n/L$ where $L$ is a lattice in $\mathbb{C}^n$, K3-surfaces, compact Calabi-Yau manifolds, quadrics, products of projective spaces and so on.

References

• Wikipedia, Fubini-Study metric