Sum Of Residues Formula

In mathematics, the residue formula says that the sum of the residues of a meromorphic differential form on a smooth proper algebraic curve vanishes.

Statement

In this article, ''X'' denotes a proper smooth algebraic curve over a field ''k''. A meromorphic (algebraic) differential form $\omega$ has, at each closed point ''x'' in ''X'', a residue which is denoted $\operatorname_x \omega$. Since $\omega$ has poles only at finitely many points, in particular the residue vanishes for all but finitely many points. The residue formula states:
:$\sum_ \operatorname_x \omega=0.$

Proofs

A geometric way of proving the theorem is by reducing the theorem to the case when ''X'' is the projective line, and proving it by explicit computations in this case, for example in .

proves the theorem using a notion of traces for certain endomorphisms of infinite-dimensional vector spaces. The residue of a differential form $f dg$ can be expressed in terms of traces of endomorphisms on the fraction field $K_x$ of the completed local rings $\hat \mathcal O_$ which leads to a conceptual proof of the formula. A more recent exposition along similar lines, using more explicitly the notion of Tate vector spaces, is given by .

References

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*{{Citation, last=Tate, first=John, title=Residues of differentials on curves, journal=Annales scientifiques de l'École Normale Supérieure, series=4, volume=1, year=1968, issue=1, pages=149-159, url=http://www.numdam.org/item/?id=ASENS_1968_4_1_1_149_0, doi=10.24033/asens.1162, doi-access=free

Algebraic geometry
Algebraic curves
Differential forms