C, the resulting spectral sequence degenerates at . So the cohomology of with complex coefficients has a decreasing filtration, the Hodge filtration, whose associated graded vector spaces are the algebraically defined groups . This is part of the mixed Hodge structure which Deligne defined on the cohomology of any complex algebraic variety. In particular, there is also a weight filtration on the rational cohomology of . The resulting filtration on can be constructed using the logarithmic de Rham complex. Namely, define an increasing filtration by The resulting filtration on cohomology is the weight filtration: Building on these results, Hélène Esnault and Eckart Viehweg generalized the Kodaira–Akizuki–Nakano vanishing theorem in terms of logarithmic differentials. Namely, let X be a smooth complex projective variety of dimension n, D a divisor with simple normal crossings on X, and L an ample line bundle on X. Then and for all . See also Adjunction formula Borel–Moore homology Differential of the first kind Log structure Mixed Hodge structure Residue theorem Poincaré residue Notes References External links Aise Johan de Jong, Algebraic de Rham cohomology. Complex analysis Algebraic geometry