Holomorphic Log Complex
By definition of and the fact that exterior differentiation d satisfies, one has
- .
This implies that there is a complex of sheaves, known as the holomorphic log complex corresponding to the divisor D. This is a subcomplex of, where is the inclusion and is the complex of sheaves of holomorphic forms on .
Of special interest is the case where D has simple normal crossings. Then if are the smooth, irreducible components of, one has with the meeting transversely. Locally is the union of hyperplanes, with local defining equations of the form in some holomorphic coordinates. One can show that the stalk of at p satisfies
and that
- .
Some authors, e.g., use the term log complex to refer to the holomorphic log complex corresponding to a divisor with normal crossings.
Read more about this topic: Logarithmic Form
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