Hodge structure
The total cohomology of a compact Kähler manifold has a mixed Hodge structure, where the "n"th space of the weight filtration "Wn" is the direct sum of the cohomology groups (with rational coefficients) of degree less than or equal to "n". Therefore, one can think of classical Hodge theory in the compact, complex case as providing a double grading on the complex cohomology group, which defines an increasing fitration "Fp" and a decreasing filtration "Wn" that are compatible in certain way. In general, the total cohomology space still has these two filtrations, but they no longer come from a direct sum decomposition. In relation with the third definition of the pure Hodge structure, one can say that a mixed Hodge structure cannot be described using the action of the group formula_57 An important insight of Deligne is that in the mixed case there is a more complicated noncommutative proalgebraic group that can be used to the same effect using Tannakian formalism.