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 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.

$\lambda (X,f,\alpha )=(X,f,\lambda \alpha )$
$(X,f,\alpha )=0$ if $\dim f(X)<k$ .
$\ \sum _{i}(X_{i},f_{i},\alpha _{i})=0$ 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},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 $\operatorname {ker} \;\partial /\operatorname {im} \;\partial$ .