Logarithmic form: Difference between revisions

Revision as of 02:37, 28 April 2023

Revision as of 18:48, 6 July 2023

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==Historical terminology==

In the 19th-century theory of [[elliptic function]]s, 1-forms with logarithmic poles were sometimes called ''integrals of the second kind'' (and, with an unfortunate inconsistency, sometimes ''differentials of the third kind''). For example, the [[Weierstrass zeta function]] associated to a [[lattice_lattice (group)|lattice]] <math>\Lambda</math> in '''C''' was called an "integral of the second kind" to mean that it could be written

:<math>\zeta(z)=\frac{\sigma'(z)}{\sigma(z)}.</math>

In modern terms, it follows that <math>\zeta(z)dz=d\sigma/\sigma</math> is a 1-form on '''C''' with logarithmic poles on <math>\Lambda</math>, since <math>\Lambda</math> is the zero set of the Weierstrass sigma function <math>\sigma(z).</math>

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==References==

* {{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Équations différentielles à points singuliers réguliers | series=Lecture Notes in Mathematics | publisher=[[Springer-Verlag]] | oclc=169357 | year=1970 | volume=163 | isbn=3540051902 | mr=0417174|doi=10.1007/BFb0061194|url=https://publications.ias.edu/node/355}}

* {{citation|author1-last=Deligne|author1-first=Pierre|author1-link=Pierre Deligne|title=Théorie de Hodge II|journal=Publ. Math. IHESIHÉS |volume=40|pages=5–57|year=1971|mr=0498551|url=http://www.numdam.org/item/PMIHES_1971__40__5_0/}}

*{{Citation|author1-last=Esnault|author1-first=Hélène | author1-link=Hélène Esnault | author2-last=Viehweg | author2-first=Eckart | author2-link=Eckart Viehweg | title=Lectures on vanishing theorems | publisher=Birkhäuser| isbn=978-3-7643-2822-1 |mr=1193913 | year=1992|doi=10.1007/978-3-0348-8600-0}}

*{{citation|last1=Griffiths |first1=Phillip | author-link1=Phillip Griffiths |last2=Harris |first2=Joseph |author-link2=Joe Harris (mathematician) | title=Principles of algebraic geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | orig-year=1978 | isbn=0-471-05059-8 | mr=0507725|doi=10.1002/9781118032527}}