 More precisely, one should be able to assign to any algebraic variety "X" a polynomial "P""X"("t"), called its virtual Poincaré polynomial, with the properties
The existence of such polynomials would follow from the existence of an analogue of Hodge structure in the cohomologies of a general (singular and non-complete) algebraic variety.The novel feature is that the "n"th cohomology of a general variety looks as if it contained pieces of different weights.This led Alexander Grothendieck to his conjectural theory of motives and motivated a search for an extension of Hodge theory, which culminated in the work of Pierre Deligne.