Editing Logarithmic form (section)

==Mixed Hodge theory for smooth varieties==
Over the complex numbers, Deligne proved a strengthening of [[Alexander Grothendieck]]'s algebraic de Rham theorem, relating [[coherent sheaf cohomology]] with [[singular cohomology]]. Namely, for any smooth scheme ''X'' over '''C''' with a divisor with simple normal crossings ''D'', there is a natural isomorphism
:<math> H^k(X, \Omega^{\bullet}_X(\log D)) \cong H^k(X-D,\mathbf{C})</math>
for each integer ''k'', where the groups on the left are defined using the [[Zariski topology]] and the groups on the right use the classical (Euclidean) topology.<ref>Deligne (1970), Corollaire II.6.10.</ref>

Moreover, when ''X'' is smooth and [[proper morphism|proper]] over '''C''', the resulting [[spectral sequence]]
:<math>E_1^{pq} = H^q(X,\Omega^p_X(\log D)) \Rightarrow H^{p+q}(X-D,\mathbf{C})</math>
degenerates at <math>E_1</math>.<ref>Deligne (1971), Corollaire 3.2.13.</ref> So the cohomology of <math>X-D</math> with complex coefficients has a decreasing filtration, the '''Hodge filtration''', whose associated graded vector spaces are the algebraically defined groups <math>H^q(X,\Omega^p_X(\log D))</math>.

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 <math>X-D</math>. The resulting filtration on <math>H^*(X-D,\mathbf{C})</math> can be constructed using the logarithmic de Rham complex. Namely, define an increasing filtration <math>W_{\bullet} \Omega^p_X(\log D) </math> by
:<math>W_{m}\Omega^p_X(\log D) =  \begin{cases}
0 & m < 0\\
\Omega^{p-m}_X\cdot \Omega^m_X(\log D) & 0\leq m \leq p\\
\Omega^p_X(\log D) & m\geq p.
\end{cases} </math>
The resulting filtration on cohomology is the weight filtration:<ref>Peters & Steenbrink (2008), Theorem 4.2.</ref>
:<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
:<math>H^q(X,\Omega^p_X(\log D)\otimes O_X(-D)\otimes L)=0</math>
for all <math>p+q>n</math>.<ref>Esnault & Viehweg (1992), Corollary 6.4.</ref>