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There is an [[exact sequence]] of [[coherent sheaves]] on ''X'':
:<math>0 \to \Omega^1_X \to \Omega^1_X(\log D) \overset{\beta}\to \oplus_j ({i_j})_*\mathcal{O}_{D_j} \to 0,</math>
where <math>i_j: D_j \to X</math> is the inclusion of an irreducible component of ''D''. Here β is called the '''residue''' map; so this sequence says that a 1-form with log poles along ''D'' is regular (that is, has no poles) [[if and only if]] its residues are zero. More generally, for any ''p'' ≥ 0, there is an exact sequence of coherent sheaves on ''X'':
: <math>0 \to \Omega^p_X \to \Omega^p_X(\log D) \overset{\beta}\to \oplus_j ({i_j})_*\Omega^{p-1}_{D_j}(\log (D-D_j)) \to \cdots \to 0,</math>
where the sums run over all irreducible components of given dimension of intersections of the divisors ''D''<sub>''j''</sub>. Here again, β is called the residue map.
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:<math> W_mH^k(X-D, \mathbf{C}) = \text{Im}(H^k(X, W_{m-k}\Omega^{\bullet}_X(\log D))\rightarrow H^k(X-D,\mathbf{C})).</math>
Building on these results, [[Hélène Esnault]] and [[Eckart Viehweg]] generalized the [[Nakano vanishing theorem|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
:<math>H^q(X,\Omega^p_X(\log D)\otimes L)=0</math>
and
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