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 $\{Bbb\; C\}$ 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.

(i) $lambda\; (X,\; f,\; alpha)=(X,\; f,\; lambdaalpha)$

(ii) $sum\_i(X\_i,f\_i,alpha\_i)=0$ provided that $sum\_if\_\{i*\}alpha\_iequiv\; 0$, 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 $cup\_i\; f\_i(X\_i)$.

(iii) $(X,f,alpha)=0$ if .

Defining the boundary operator

The boundary operator $partial:;\; C\_k\; mapsto\; C\_\{k-1\}$ is defined by

:$partial(X,f,alpha)=2pi\; sqrt\{-1\}sum\_i(V\_i,\; f\_i,\; res\_\{V\_i\},alpha)$,

where $V\_i$ are components of the polar divisor of $alpha$, "res" is the Poincare 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 $operatorname\{ker\};\; partial\; /\; operatorname\{im\}\; ;\; partial$.

Notes

* B. Khesin, A. Rosly, " [http://arxiv.org/abs/math/0102152 Polar Homology and Holomorphic Bundles] " Phil. Trans. Roy. Soc. Lond. A359 (2001) 1413-1428