Polar homology

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 $$C_k of polar "k"-chains is a vector space over $${Bbb C} defined as a quotient $$A_k/R_k, with $$A_k and $$R_k vector spaces defined below.

Defining $$A_k

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

Defining $$R_k

The space $$R_k is generated by the following relations.

(i) $$lambda (X, f, alpha)=(X, f, lambdaalpha)

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

(iii) $$(X,f,alpha)=0 if $$dim f(X) < k.

Defining the boundary operator

The boundary operator $$partial:; C_k mapsto C_{k-1} is defined by

:$$partial(X,f,alpha)=2pi sqrt{-1}sum_i(V_i, f_i, res_{V_i},alpha),

where $$V_i are components of the polar divisor of $$alpha, "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.

Notes

* B. Khesin, A. Rosly, " [http://arxiv.org/abs/math/0102152 Polar Homology and Holomorphic Bundles] " Phil. Trans. Roy. Soc. Lond. A359 (2001) 1413-1428