Given a convex polytope whose facets are simplices, define the f-vector by f_i = the number of i-dim faces. Which vectors of integers are f-vectors? A list of conditions was conjectured, proven sufficients by direct construction of enough polytopes, and proven necessary by applying hard Lefschetz to the (rationally smooth) toric variety associated to the dual polytope. (A combinatorial proof came later.) See Fulton's book on toric varieties.