Polar homology
In complex geometry, a polar homology is a group which captures holomorphic invariants of a complex manifold in a similar way to usual homology of a manifold in differential topology. Polar homology was defined by B. Khesin and A. Rosly in 1999. Definition. 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. Defining formula_4. 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. Defining formula_5. The space formula_5 is generated by the following relations. Defining the boundary operator. The boundary operator formula_22 is defined by where formula_24 are components of the polar divisor of formula_10, "res" is the Poincaré residue, and formula_26 are restrictions of the map "f" to each component of the divisor. Khesin and Rosly proved that this boundary operator is well defined, and satisfies formula_27. They defined the polar cohomology as the quotient formula_28.