The hard Lefschetz theorem is a theorem in complex differential geometry, more specifically Kahler geometry, which determines an isomorphisms between certain cohomology spaces. Together with the Poincare duality, Serre duality it shows just how rigid the cohomology groups of a compact Kahler manifold are. In this post the theorem and its proof will be stated, with a view towards the more rudimentary setting of symplectic geometry.
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