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Polar homology

From Wikipedia, the free encyclopedia

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. if .
  2. 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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