Kähler manifold
where formula_15 is the Levi-Civita connection of formula_16. In this case, formula_17 is called a "Kähler structure", and a "Kähler manifold" is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties.