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

Defining [math]\displaystyle{ A_k }[/math]

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

Defining [math]\displaystyle{ R_k }[/math]

The space [math]\displaystyle{ R_k }[/math] is generated by the following relations.

1. [math]\displaystyle{ \lambda (X, f, \alpha)=(X, f, \lambda\alpha) }[/math]
2. [math]\displaystyle{ (X,f,\alpha)=0 }[/math] if [math]\displaystyle{ \dim f(X) \lt k }[/math].
3. [math]\displaystyle{ \ \sum_i(X_i,f_i,\alpha_i)=0 }[/math] provided that

[math]\displaystyle{ \sum_if_{i*}\alpha_i\equiv 0, }[/math]

where

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

Defining the boundary operator

The boundary operator [math]\displaystyle{ \partial:\; C_k \mapsto C_{k-1} }[/math] is defined by

[math]\displaystyle{ \partial(X,f,\alpha)=2\pi \sqrt{-1}\sum_i(V_i, f_i, res_{V_i}\,\alpha) }[/math],

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

Notes

• B. Khesin, A. Rosly, Polar Homology and Holomorphic Bundles Phil. Trans. Roy. Soc. Lond. A359 (2001) 1413-1428