The Kodaira embedding theorem characterizes smooth complex projective varieties among all compact Kähler manifolds. Namely, a compact complex manifold "X" is projective if and only if there is a Kähler form "ω" on "X" whose class in is in the image of the integral cohomology group . (Because a positive multiple of a Kähler form is a Kähler form, it is equivalent to say that "X" has a Kähler form whose class in is in .) Equivalently, "X" is projective if and only if there is a holomorphic line bundle "L" on "X" with a hermitian metric whose curvature form ω is positive (since ω is then a Kähler form that represents the first Chern class of "L" in ).