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  "body": "\n## Summary\n\nA Hermitian manifold is a complex manifold with a Riemannian metric that is compatible with multiplication by $i$ on tangent vectors.\nIt is the complex analogue of a Riemannian manifold, and every complex manifold admits such a metric.\nEach Hermitian metric has a fundamental $2$-form, and when this form is closed the manifold is a Kähler manifold.\nKähler manifolds are the setting of Hodge theory and of much of complex algebraic geometry.\n\n## Explanations\n\n### L1 human\n\nOn a complex manifold one can turn each tangent vector by a quarter turn, using multiplication by $i$.\nA Hermitian metric measures lengths and angles so that this quarter turn never changes a length or an angle.\nThe plane $\\C$ with the usual distance is the simplest example: rotating an arrow by $90$ degrees keeps its length.\nPairing the metric with the quarter turn gives a $2$-form, which measures oriented areas.\n",
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