 Let "M" be a complex projective manifold.The space formula_1 of polar "k"-chains is a vector space over formula_2 defined as a quotient formula_3, with formula_4 and formula_5 vector spaces defined below.The space formula_4 is freely generated by the triples formula_8, where "X" is a smooth, "k"-dimensional complex manifold, formula_9 a holomorphic map, and formula_10 is a rational "k"-form on "X", with first order poles on a divisor with normal crossing.