Khesin, Boris; Rosly, Alexei Polar linkings, intersections and Weil pairing. (English) Zbl 1222.14039 Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2063, 3505-3524 (2005). Summary: 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. Cited in 1 Document MSC: 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14C25 Algebraic cycles 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 32Q25 Calabi-Yau theory (complex-analytic aspects) Keywords:linking number; Poincaré residue; polar homology; Parshin symbols; meromorphic forms PDFBibTeX XMLCite \textit{B. Khesin} and \textit{A. Rosly}, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2063, 3505--3524 (2005; Zbl 1222.14039) Full Text: DOI Link References: [1] Atiyah, M.F. 1981 Green’s functions for self-dual four-manifolds. <i>Advances in mathematical supplement studies</i>, vol. 7A. pp. 129–158, New York: Academic Press [2] Brylinski, J.-L. & McLaughlin, D.A. 1996 The geometry of two-dimensional symbols. <i>K-Theory</i> <b>10</b>, 215–237, (doi:10.1007/BF00538183). · Zbl 0870.32004 [3] Deligne, P. 1971 Théorie de Hodge. II. <i>Inst. Hautes Études Sci. Publ. Math.</i> <b>40</b>, 5–57. · Zbl 0219.14007 [4] Donaldson, S.K. & Thomas, R.P. 1998 Gauge theory in higher dimensions. <i>The geometric universe (Oxford, 1996)</i>. Oxford: Oxford University Press, pp. 31–47. · Zbl 0926.58003 [5] Frenkel, I.B. & Khesin, B.A. 1996 Four dimensional realization of two-dimensional current groups. <i>Commun. Math. Phys.</i> <b>178</b>, 541–561. [6] Frenkel, I. B. & Todorov, A. N. 2002 Complex counterpart of Chern–Simons–Witten theory and holomorphic linking. Preprint. · Zbl 1135.32021 [7] Griffiths, P.A. & Harris, J. 1978 Principles of algebraic geometry. New York: Wiley. · Zbl 0408.14001 [8] Khesin, B. & Rosly, A. 2001 Polar homology and holomorphic bundles. <i>Phil. Trans. R. Soc. A</i> <b>359</b>, 1413–1427, (doi:10.1098/rsta.2001.0844). · Zbl 0996.32012 [9] Khesin, B. & Rosly, A. 2003 Polar homology. <i>Can. J. Math.</i> <b>55</b>, 1100–1120, (http://arxiv.org/absmath.AG/009015). · Zbl 1041.32016 [10] Khesin, B., Rosly, A. & Thomas, R. 2004 Polar de Rham theorem. <i>Topology</i> <b>43</b>, 1231–1246, (http://arxiv.org/absmath.AG/0305081), (doi:10.1016/j.top.2004.01.001). · Zbl 1060.14013 [11] Parshin, A.N. 1977 Residues and duality on algebraic surfaces. <i>Uspehi Math. Nauk</i> <b>32</b>, 225–226. · Zbl 0358.14013 [12] Polyakov, A.M. 1988 Fermi–Bose transmutations induced by gauge fields. <i>Modern Phys. Lett. A</i> <b>3</b>, 325–328, (doi:10.1142/S0217732388000398). [13] Schwarz, A.S. 1977/78 The partition function of degenerate quadratic functional and Ray–Singer invariants. <i>Lett. Math. Phys.</i> <b>2</b>, 247–252, (doi:10.1007/BF00406412). · Zbl 0383.70017 [14] Thomas, R. P. 1997 Gauge theory on Calabi–Yau manifolds. Ph.D thesis, University of Oxford. [15] Viro, O. 2001 Encomplexing the writhe, topology, ergodic theory, real algebraic geometry. <i>Am. Math. Soc. Transl. Ser. 2</i> <b>202</b>, 241–256. · Zbl 0983.57006 [16] Witten, E. 1989 Quantum field theory and the Jones polynomial. <i>Commun. Math. Phys.</i> <b>121</b>, 351–399, (doi:10.1007/BF01217730). [17] Witten, E. 1995 Chern–Simons gauge theory as a string theory. <i>The Floer memorial volume, Progress in mathematics</i>, vol. 133. Basel: Birkhäuser, pp. 637–678. (http://arxiv.org/abshep-th/9207094). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.