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 $$C_k$$ of polar k-chains is a vector space over $${\mathbb C}$$ defined as a quotient $$A_k/R_k$$, with $$A_k$$ and $$R_k$$ vector spaces defined below.

Defining Ak
The space $$A_k$$ is freely generated by the triples $$(X, f, \alpha)$$, where X is a smooth, k-dimensional complex manifold, $$f:\; X \mapsto M$$ a holomorphic map, and $$\alpha$$ is a rational k-form on X, with first order poles on a divisor with normal crossing.

Defining Rk
The space $$R_k$$ is generated by the following relations.


 * 1) $$\lambda (X, f, \alpha)=(X, f, \lambda\alpha)$$
 * 2) $$(X,f,\alpha)=0$$ if $$\dim f(X) < k$$.
 * 3) $$\ \sum_i(X_i,f_i,\alpha_i)=0$$ provided that
 * $$\sum_if_{i*}\alpha_i\equiv 0,$$


 * where


 * $$dim \;f_i(X_i)=k$$ for all $$i$$ and the push-forwards $$f_{i*}\alpha_i$$ are considered on the smooth part of $$\cup_i f_i(X_i)$$.

Defining the boundary operator
The boundary operator $$\partial:\; C_k \mapsto C_{k-1}$$ is defined by


 * $$\partial(X,f,\alpha)=2\pi \sqrt{-1}\sum_i(V_i, f_i, res_{V_i}\,\alpha)$$,

where $$V_i$$ are components of the polar divisor of $$\alpha$$, res is the Poincaré residue, and $$f_i=f|_{V_i}$$ 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 $$\partial^2=0$$. They defined the polar cohomology as the quotient $$ \operatorname{ker}\; \partial / \operatorname{im} \; \partial$$.