---
tag: NT4797Z9H8
kind: concept
name: Hermitian symmetric space
msc: [32M15, 53C35]
specializes: [PAA93R4WYD, CK1DCPC3D3]
sources:
  - url: https://en.wikipedia.org/wiki/Hermitian_symmetric_space
same_as:
  wikidata: Q5741926
  wikipedia: Hermitian symmetric space
---

## Summary

A Hermitian symmetric space is a [[symmetric space|PAA93R4WYD]] with a compatible complex structure, for which every geodesic symmetry is holomorphic.
Examples are the complex projective spaces, the Grassmannians, the unit disc and the Siegel upper half space.
Such a space is automatically a Kähler manifold.
The irreducible ones of noncompact type are bounded symmetric domains in $\C^n$, and they are the home of automorphic forms in several variables.

## Explanations

### L1 human

Take a symmetric space, where each point has a mirror symmetry through it.
Suppose that the space also has complex coordinates, and that all these symmetries respect them.
Then it is a Hermitian symmetric space.
The unit disc with its hyperbolic metric is the main example: the symmetry through $0$ is $z \mapsto -z$, which is holomorphic.
The upper half plane, where modular forms live, is the same space in other coordinates.