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

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Let M be a complex projective manifold. The space   of polar k-chains is a vector space over   defined as a quotient  , with   and   vector spaces defined below.

Defining Ak

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The space   is freely generated by the triples  , where X is a smooth, k-dimensional complex manifold,   a holomorphic map, and   is a rational k-form on X, with first order poles on a divisor with normal crossing.

Defining Rk

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The space   is generated by the following relations.

  1.  
  2.   if  .
  3.   provided that
 
where
  for all   and the push-forwards   are considered on the smooth part of  .

Defining the boundary operator

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The boundary operator   is defined by

 ,

where   are components of the polar divisor of  , res is the Poincaré residue, and   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  . They defined the polar cohomology as the quotient  .

Notes

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