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 $ℂ$ defined as a quotient $A_k/R_k$, with $A_k$ and $R_k$ vector spaces defined below.

Defining A_k

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 R_k

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

$\sum _i{f}_{i*}{\alpha }_i\equiv 0,$

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

$\partial (X,f,\alpha )=2\pi \sqrt{-1}\sum _i(V_i,f_i,re{s}_{V_i}\alpha )$,

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 $\ker \partial /\mathrm{im}\partial$.

Notes

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

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