Polar homology and holomorphic bundles

B. Khesin, A. Rosly

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9 Citations (Scopus)


We describe polar homology groups for complex manifolds. The polar κ-chains are subvarieties of complex dimension κ with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincaré 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.

Original languageEnglish
Pages (from-to)1413-1427
Number of pages15
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number1784
Publication statusPublished - 15 Jul 2001
Externally publishedYes


  • Complex manifold
  • Divisor of poles
  • Gauge transformations
  • Poincaré residue
  • Poisson structure


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