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Classically, for example in [[elliptic function]] theory, the logarithmic differential forms were recognised as complementary to the [[differentials of the first kind]]. They were sometimes called ''differentials of the second kind'' (and, with an unfortunate inconsistency, also sometimes ''of the third kind''). The classical theory has now been subsumed as an aspect of Hodge theory. For a Riemann surface ''S'', for example, the differentials of the first kind account for the term ''H''<sup>1,0</sup> in ''H''<sup>1</sup>(''S''), when by the [[Dolbeault isomorphism]] it is interpreted as the [[sheaf cohomology]] group ''H''<sup>0</sup>(''S'',Ω); this is tautologous considering their definition. The ''H''<sup>1,0</sup> direct summand in ''H''<sup>1</sup>(''S''), as well as being interpreted as ''H''<sup>1</sup>(''S'',O) where O is the sheaf of [[holomorphic function]]s on ''S'', can be identified more concretely with a vector space of logarithmic differentials.
== Sheaf of logarithmic forms ==
In [[algebraic geometry]], the [[sheaf (mathematics)|sheaf]] of '''logarithmic differential ''p''-forms''' <math>\Omega^p_X(\log D)</math> on a [[Smooth scheme|smooth]] [[projective variety]] ''X'' along a smooth [[divisor (algebraic geometry)|divisor]] <math>D = \sum D_j</math> is defined and fits into the [[exact sequence]] of locally free sheaves:
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