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Wavelength (talk | contribs) inserting 2 hyphens: —> "higher-dimensional" and "Higher-dimensional"—User talk:Wavelength#Hyphenation [to Archive 6] |
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This implies that there is a complex of sheaves <math>( \Omega^{\bullet}_X(\log D), d) </math>, known as the ''holomorphic log complex'' corresponding to the divisor ''D''. This is a subcomplex of <math> j_*\Omega^{\bullet}_{X-D} </math>, where <math> j:X-D\rightarrow X </math> is the inclusion and <math> \Omega^{\bullet}_{X-D} </math> is the complex of sheaves of holomorphic forms on ''X''−''D''.
Of special interest is the case where ''D'' has simple [[normal crossings]]. Then if <math> \{D_{\nu}\} </math> are the smooth, irreducible components of ''D'', one has <math> D = \sum D_{\nu} </math> with the <math> D_{\nu} </math> meeting transversely. Locally ''D'' is the union of hyperplanes, with local defining equations of the form <math> z_1\cdots z_k = 0 </math> in some holomorphic coordinates. One can show that the stalk of <math> \Omega^1_X(\log D) </math> at ''p'' satisfies<ref name="foo">Chris A.M. Peters; Joseph H.M. Steenbrink (2007). Mixed Hodge Structures. Springer. ISBN 978-3-540-
:<math>\Omega_X^1(\log D)_p = \mathcal{O}_{X,p}\frac{dz_1}{z_1}\oplus\cdots\oplus\mathcal{O}_{X,p}\frac{dz_k}{z_k} \oplus \mathcal{O}_{X,p}dz_{k+1} \oplus \cdots \oplus \mathcal{O}_{X,p}dz_n</math>
and that
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