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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 ${\displaystyle C_{k}}$ of polar k-chains is a vector space over ${\displaystyle {\mathbb {C} }}$ defined as a quotient ${\displaystyle A_{k}/R_{k}}$, with ${\displaystyle A_{k}}$ and ${\displaystyle R_{k}}$ vector spaces defined below.

Defining A_k

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

Defining R_k

The space ${\displaystyle R_{k}}$ is generated by the following relations.

1. ${\displaystyle \lambda (X,f,\alpha )=(X,f,\lambda \alpha )}$
2. ${\displaystyle (X,f,\alpha )=0}$ if ${\displaystyle \dim f(X)<k}$.
3. ${\displaystyle \ \sum _{i}(X_{i},f_{i},\alpha _{i})=0}$ provided that

${\displaystyle \sum _{i}f_{i*}\alpha _{i}\equiv 0,}$

where

${\displaystyle dim\;f_{i}(X_{i})=k}$ for all ${\displaystyle i}$ and the push-forwards ${\displaystyle f_{i*}\alpha _{i}}$ are considered on the smooth part of ${\displaystyle \cup _{i}f_{i}(X_{i})}$.

Defining the boundary operator

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

${\displaystyle \partial (X,f,\alpha )=2\pi {\sqrt {-1}}\sum _{i}(V_{i},f_{i},res_{V_{i}}\,\alpha )}$,

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

Notes

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

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