a Kähler manifold
/ KAY-ler /
A complex manifold carries a complex structure J; a Riemannian manifold carries a metric g for measuring lengths; a symplectic manifold carries a closed 2-form omega for measuring areas in phase space. A Kähler manifold is the rare and beautiful situation where all three structures live together and are perfectly compatible — the metric, the complex structure, and the symplectic form lock into one another so tightly that complex geometry, Riemannian geometry, and symplectic geometry all describe the same object at once. This triple coincidence is why Kähler manifolds are the central characters of complex geometry and the natural home of algebraic varieties.
Precisely, start with a Hermitian metric on a complex manifold: a Riemannian metric g such that g(JX, JY) = g(X, Y). From g and J build the fundamental 2-form omega(X, Y) = g(JX, Y), called the Kähler form (a (1,1)-form). The metric is Kähler if and only if this form is closed: d omega = 0. That one differential equation is equivalent to a whole list of conditions, any of which can be taken as the definition: the complex structure J is parallel with respect to the Levi-Civita connection; the Levi-Civita connection equals the Chern connection; locally the metric is given by a single real function K (the Kähler potential) via g_{i,j-bar} = (second mixed derivative of K), i.e. omega = i partial partial-bar K. The condition d omega = 0 means omega is also a symplectic form, so a Kähler manifold is simultaneously symplectic.
Compact Kähler manifolds enjoy a cascade of powerful theorems unavailable in general: the Hodge decomposition (cohomology splits by (p,q)-type), Hodge symmetry h^{p,q} = h^{q,p}, the hard Lefschetz theorem, the partial partial-bar lemma, and via Kodaira, projective embeddability when a positive line bundle exists. Every smooth projective variety is Kähler (restrict the Fubini-Study metric of projective space). The crucial honesty point: not every complex manifold is Kähler. Compact complex manifolds with odd first Betti number (Hopf surfaces, most non-algebraic surfaces) cannot be Kähler, because Hodge symmetry would force b_1 to be even. Being complex, or even being symplectic, does not give you Kähler for free.
Complex projective space CP^n with the Fubini-Study metric is the model Kähler manifold: its Kähler form is omega = (i/2) partial partial-bar log(1 + |z|^2) in an affine chart, which is closed and positive. Because every smooth projective variety inherits this metric by restriction, all of them are Kähler — which is why algebraic geometry and Kähler geometry overlap so heavily.
CP^n with the Fubini-Study metric is Kähler, so every smooth projective variety is too.
Complex does not imply Kähler. Any compact complex manifold with odd first Betti number (such as a Hopf surface) fails to be Kähler, since Hodge symmetry on a compact Kähler manifold forces b_1 to be even.