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 {\displaystyle C_{k}} of polar k-chains is a vector space over C {\displaystyle {\mathbb {C} }} defined as a quotient A k / R k {\displaystyle A_{k}/R_{k}} , with A k {\displaystyle A_{k}} and R k {\displaystyle R_{k}} vector spaces defined below.
Defining Ak The space A k {\displaystyle A_{k}} is freely generated by the triples ( X , f , α ) {\displaystyle (X,f,\alpha )} , where X is a smooth, k-dimensional complex manifold, f : X ↦ M {\displaystyle f:\;X\mapsto M} a holomorphic map, and α {\displaystyle \alpha } is a rational k-form on X, with first order poles on a divisor with normal crossing.
Defining Rk The space R k {\displaystyle R_{k}} is generated by the following relations.
λ ( X , f , α ) = ( X , f , λ α ) {\displaystyle \lambda (X,f,\alpha )=(X,f,\lambda \alpha )}
( X , f , α ) = 0 {\displaystyle (X,f,\alpha )=0} if dim f ( X ) < k {\displaystyle \dim f(X)<k} .
∑ i ( X i , f i , α i ) = 0 {\displaystyle \ \sum _{i}(X_{i},f_{i},\alpha _{i})=0} provided that
∑ i f i ∗ α i ≡ 0 , {\displaystyle \sum _{i}f_{i*}\alpha _{i}\equiv 0,}
where
d i m f i ( X i ) = k {\displaystyle dim\;f_{i}(X_{i})=k} for all i {\displaystyle i} and the push-forwards f i ∗ α i {\displaystyle f_{i*}\alpha _{i}} are considered on the smooth part of ∪ i f i ( X i ) {\displaystyle \cup _{i}f_{i}(X_{i})} .
Defining the boundary operator The boundary operator ∂ : C k ↦ C k − 1 {\displaystyle \partial :\;C_{k}\mapsto C_{k-1}} is defined by
∂ ( X , f , α ) = 2 π − 1 ∑ i ( V i , f i , r e s V i α ) {\displaystyle \partial (X,f,\alpha )=2\pi {\sqrt {-1}}\sum _{i}(V_{i},f_{i},res_{V_{i}}\,\alpha )} , where V i {\displaystyle V_{i}} are components of the polar divisor of α {\displaystyle \alpha } , res is the Poincaré residue, and f i = f | V i {\displaystyle 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 {\displaystyle \partial ^{2}=0} . They defined the polar cohomology as the quotient ker ∂ / im ∂ {\displaystyle \operatorname {ker} \;\partial /\operatorname {im} \;\partial } .
Notes B. Khesin, A. Rosly, Polar Homology and Holomorphic Bundles Phil. Trans. Roy. Soc. Lond. A359 (2001) 1413-1428