Polar Homology and Holomorphic Bundles
arXiv:math/0102152 · doi:10.1098/rsta.2001.0844
Abstract
We describe polar homology groups for complex 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 Poincare residue on it. The polar homology groups may be regarded as holomorphic analogues of the homology groups in topology. We also describe the polar homology groups for quasi-projective one-dimensional varieties (affine curves). These groups obey the Mayer--Vietoris property. A complex counterpart of the Gauss linking number of two curves in a three-fold and various gauge-theoretic aspects of the above correspondence are also discussed.
16 pages; expanded version of the talk given by one of the authors at the discussion meeting "Topological methods in the physical sciences" London, Nov. 15-16, 2000; to appear in Philos. Trans. Roy. Soc. London Ser. A