## Abstract

Polar homology and linkings arise as natural holomorphic analogues in algebraic geometry of the homology groups and links in topology. For complex projective manifolds, the polar *k*-chains are subvarieties of complex dimension *k* with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincaré residue on it. We also define the corresponding analogues for the intersection and linking numbers of complex submanifolds, and show that they have properties similar to those of the corresponding topological objects. Finally, we establish the relation between the holomorphic linking and the Weil pairing of functions on a complex curve and its higher-dimensional counterparts.

## Footnotes

↵Note that the consideration of triples (

*A*,*f*,*α*) instead of pairs (), which we used in §1, is similar to the definition of chains in the singular homology theory. In the latter case, although one considers the mappings of abstract simplices into the manifold, morally, it is only ‘images of simplices’ that matter. Here lies an important distinction; unlike the topological homology, where in each dimension*k*, one uses all continuous maps of one standard object (the standard*k*-simplex or the standard*k*-cell) to a given topological space, in polar homology, we deal with complex analytic maps of a large class of*k*-dimensional varieties to a given one.↵An example of the polar divisor {

*xy*=0} for the form d*x*∧d*y*/*xy*in should be viewed as a complexification of a polygon vertex in . Indeed, the cancellation of the repeated residues on different components of the divisor is mimicking the calculation of the boundary of a boundary of a polygon; every polygon vertex appears twice with different signs as a boundary point of two sides.↵Note that the polar homology is an analogue of singular homology with coefficients in ℝ or ℂ.

↵An important property of such forms on projective varieties is that they are closed, see Deligne (1971).

↵For instance, on a complex curve

*X*of genus*g*one has , and a holomorphic 1-differential representing a generic element in HP_{1}(*X*) has 2*g*−2 zeros. From this point of view, the complex genus*g*curve is like a graph that has*g*loops joined by*g*−1 edges and having 2*g*−2 trivalent (i.e. ‘non-smooth’) points. The ‘smooth orientable cases’ are , which corresponds to a real segment, and an elliptic curve, which is a complex counterpart of the circle in this precise sense.- Received January 4, 2005.
- Accepted April 14, 2005.

- © 2005 The Royal Society

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