Hodge structure
may be much smaller than the whole group formula_44torsion, even if the Hodge number formula_45 is big. In short, the Hodge conjecture predicts that the possible "shapes" of complex subvarieties of "X" (as described by cohomology) are determined by the Hodge structure of "X" (the combination of integral cohomology with the Hodge decomposition of complex cohomology).