Hodge theory
The "Kähler package" is a powerful set of restrictions on the cohomology of smooth complex projective varieties (or compact Kähler manifolds), building on Hodge theory. The results include the Lefschetz hyperplane theorem, the hard Lefschetz theorem, and the Hodge-Riemann bilinear relations. Hodge theory and extensions such as non-abelian Hodge theory also give strong restrictions on the possible fundamental groups of compact Kähler manifolds.