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  "body": "\n## Summary\n\nA Hermitian symmetric space is a [[symmetric space|PAA93R4WYD]] with a compatible complex structure, for which every geodesic symmetry is holomorphic.\nExamples are the complex projective spaces, the Grassmannians, the unit disc and the Siegel upper half space.\nSuch a space is automatically a Kähler manifold.\nThe irreducible ones of noncompact type are bounded symmetric domains in $\\C^n$, and they are the home of automorphic forms in several variables.\n\n## Explanations\n\n### L1 human\n\nTake a symmetric space, where each point has a mirror symmetry through it.\nSuppose that the space also has complex coordinates, and that all these symmetries respect them.\nThen it is a Hermitian symmetric space.\nThe unit disc with its hyperbolic metric is the main example: the symmetry through $0$ is $z \\mapsto -z$, which is holomorphic.\nThe upper half plane, where modular forms live, is the same space in other coordinates.\n",
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