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Wavelength (talk | contribs) revising letter case—MOS:HEAD |
Wavelength (talk | contribs) inserting 2 hyphens: —> "higher-dimensional" and "Higher-dimensional"—User talk:Wavelength#Hyphenation [to Archive 6] |
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:<math>\omega = \frac{df}{f} =\left(\frac{m}{z} + \frac{g'(z)}{g(z)}\right)dz</math>
for some [[meromorphic function]] (resp. [[rational function]]) <math> f(z) = z^mg(z) </math>, where ''g'' is holomorphic and non-vanishing at 0, and ''m'' is the order of ''f'' at ''0''.. That is, for some [[open covering]], there are local representations of this differential form as a [[logarithmic derivative]] (modified slightly with the [[exterior derivative]] ''d'' in place of the usual [[differential operator]] ''d/dz''). Observe that ω has only simple poles with integer residues. On higher
==Holomorphic log complex==
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Some authors, e.g.,<ref name = "foo2">Phillip A. Griffiths; Joseph Harris (1979). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.</ref> use the term ''log complex'' to refer to the holomorphic log complex corresponding to a divisor with normal crossings.
===Higher
Consider a once-punctured elliptic curve, given as the locus ''D'' of complex points (''x'',''y'') satisfying <math> g(x,y) = y^2 - f(x) = 0 </math>, where <math>f(x) = x(x-1)(x-\lambda) </math> and <math> \lambda\neq 0,1 </math> is a complex number. Then ''D'' is a smooth irreducible [[hypersurface]] in '''C'''<sup>2</sup> and, in particular, a divisor with simple normal crossings. There is a meromorphic two-form on '''C'''<sup>2</sup>
:<math> \omega =\frac{dx\wedge dy}{g(x,y)} </math>
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