Samiksha Jaiswal (Editor)

Polar homology

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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.

Contents

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 {\displaystyle A_{k}}

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

Defining R k {\displaystyle R_{k}}

The space R k is generated by the following relations.

  1. λ ( X , f , α ) = ( X , f , λ α )
  2. ( X , f , α ) = 0 if dim ⁡ f ( X ) < k .
  3.   ∑ i ( X i , f i , α i ) = 0 provided that
where d i m f i ( X i ) = k for all i and the push-forwards f i ∗ α i are considered on the smooth part of ∪ i f i ( X i ) .

Defining the boundary operator

The boundary operator ∂ : C k ↦ C k − 1 is defined by

∂ ( X , f , α ) = 2 π − 1 ∑ i ( V i , f i , r e s V i α ) ,

where V i are components of the polar divisor of α , 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 ∂ 2 = 0 . They defined the polar cohomology as the quotient ker ∂ / im ∂ .

References

Polar homology Wikipedia
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