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.

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[edit] Definition

Let M be a complex projective manifold. The space Ck of polar k-chains is a vector space over {\Bbb C} defined as a quotient Ak / Rk, with Ak and Rk vector spaces defined below.

[edit] Defining Ak

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

[edit] Defining Rk

The space Rk is generated by the following relations.

(i) λ(X,f,α) = (X,f,λα)

(ii) \ \sum_i(X_i,f_i,\alpha_i)=0 provided that \sum_if_{i*}\alpha_i\equiv 0, where dim \;f_i(X_i)=k for all i and the push-forwards fi * αi are considered on the smooth part of \cup_i f_i(X_i).

(iii) (X,f,α) = 0 if \dim f(X) < k.

[edit] 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 Vi are components of the polar divisor of α, res is the Poincare 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.

[edit] Notes


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