describes the holomorphic vector bundle on . Then, for any , the vector bundle is the kth exterior power, The logarithmic tangent bundle means the dual vector bundle to . Explicitly, a section of is a holomorphic vector field on X that is tangent to D at all smooth points of D. Logarithmic differentials and singular cohomology Let X be a complex manifold and D a divisor with normal crossings on X. Deligne proved a holomorphic analog of de Rham's theorem in terms of logarithmic differentials. Namely, where the left side denotes the cohomology of X with coefficients in a complex of sheaves, sometimes called hypercohomology. This follows from the natural inclusion of complexes of sheaves being a quasi-isomorphism. Logarithmic differentials in algebraic geometry In algebraic geometry, the vector bundle of logarithmic differential p-forms on a smooth scheme X over a field, with respect to a divisor with simple normal crossings, is defined as above: sections of are (algebraic) differential forms ω on such that both ω and dω have a pole of order at most one along D. Explicitly, for a closed point x that lies in for and not in for , let be regular functions on some open neighborhood U of x such that is the closed subscheme defined by inside U for , and x is the closed subscheme of U defined by . Then a basis of sections of on U is given by: This describes the vector bundle on X, and then is the pth exterior power of . There is an exact sequence of coherent sheaves on X: where is the inclusion of an irreducible component of D. Here β is called the residue map; so this sequence says that a 1-form with log poles along D is regular (that is, has no poles) if and only if its residues are zero. More generally, for any p ≥ 0, there is an exact sequence of coherent sheaves on X: where the sums run over all irreducible components of given dimension of intersections of the divisors Dj. Here again, β is called the residue map. Explicitly, on an open subset of that only meets one component of , with locally defined by , the residue of a logarithmic -form along is determined by: the residue of a regular p-form is zero, whereas for any regular -form . Some authors define the residue by saying