Brylinski, J.-L.; McLaughlin, D. A. The geometry of two-dimensional symbols. (English) Zbl 0870.32004 \(K\)-Theory 10, No. 3, 215-237 (1996). The authors’ abstract: “We give a complex-analytic construction of the two-dimensional symbol of Parshin and Kato. Our approach leads to a new analytic formula for the symbol together with a direct proof of the reciprocity laws it satisfies”. Reviewer: N.Mihalache (Bucureşti) Cited in 7 Documents MSC: 32C35 Analytic sheaves and cohomology groups 32A20 Meromorphic functions of several complex variables Keywords:gerbes; two-dimensional symbols; reciprocity laws PDFBibTeX XMLCite \textit{J. L. Brylinski} and \textit{D. A. McLaughlin}, \(K\)-Theory 10, No. 3, 215--237 (1996; Zbl 0870.32004) Full Text: DOI References: [1] A’Campo, N.: Sur la monodromie des singularités isolées d’hypersurface complexes, Invent. Math. 20 (1973), 147-169. · Zbl 0264.14002 [2] Beilinson, A.: Higher regulators and values of L-functions, J. Soviet Math. 30 (1985), 2036-2070. · Zbl 0588.14013 [3] Bloch, S.: Deligne groups, unpublished manuscript (1972). [4] Bloch, S.: The Dilogarithm and Extensions of Lie Algebras, Lecture Notes in Math. 854, Springer-Verlag, New York, 1981, pp. 1-23. · Zbl 0469.14009 [5] Bloch S. and Ogus, A.: Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974), 181-202. · Zbl 0307.14008 [6] Breen, L.: On the classification of 2-gerbes and 2-stacks, Astérisque 225 (1994). · Zbl 0818.18005 [7] Brylinski, J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Math. 170, Birkhauser, Basel, 1993. · Zbl 0823.55002 [8] Brylinski, J.-L.: Holomorphic gerbes and the Beilinson regulator, K-Theory (Strasbourg 1992), Astérisque 226 (1994), 145-174. · Zbl 0821.19001 [9] Brylinski, J.-L. and McLaughlin, D. A.: The geometry of degree four characteristic classes and of line bundles on loop spaces I, Duke Math. J. 75 (3) (1994), 603-638. · Zbl 0844.57025 [10] Brylinski, J.-L. and McLaughlin, D. A.: The geometry of degree four characteristic classes and of line bundles on loop spaces II, Preprint 1994, to appear in Duke Math. J. · Zbl 0844.57025 [11] Brylinski, J.-L. and McLaughlin, D. A.: Multidimensional reciprocity laws, Preprint 1995. [12] Deligne, P.: Le symbole modéré, Publ. Math. IHES 73 (1991), 147-181. · Zbl 0749.14011 [13] Esnault, H. and Viehweg, E.: Deligne-Beilinson cohomology, in: Beilinson ’s Conjectures on Special Values of L-functions, Perspectives in Math., Academic Press, New York, 1988. · Zbl 0656.14012 [14] Gawedzki, K.: Topological actions in two-dimensional quantum field theories, in: G. Hooft, A. Jaffe, G. Mack, P. K. Mitter, and R. Stora (eds), Nonperturbative Quantum Field Theories, NATO Adv. Sci. Inst. Ser. B Physics 185, Plenum Press, New York, 1988, 101-142. [15] Giraud, J.: Cohomologie non-abélienne, Ergeb. Math. (3) 64, Springer-Verlag, Berlin, 1971. · Zbl 0226.14011 [16] Kato, K.: Milnor K-Theory and the Chow Group of Zero Cycles, Contemp. Math. 55, Amer. Math. Soc., Providence, 1986, pp. 241-253. · Zbl 0603.14009 [17] Mazur, B. and Messing, W.: Universal Extensions and One-Dimensional Crystalline Cohomology, Lecture Notes in Math. 370, Springer-Verlag, New York, 1974. · Zbl 0301.14016 [18] Milnor, J.: Singular Points of Complex Hypersurfaces, Ann. of Math. Studies 61, Princeton Univ. Press, 1968. · Zbl 0184.48405 [19] Milnor, J.: Introduction to Algebraic K-Theory, Ann. of Math. Studies 72, Princeton Univ. Press, 1971. · Zbl 0237.18005 [20] Parshin, A. N.: Local class field theory, Proc. Steklov Inst. Math. No. 3 (1985), pp. 157-185. · Zbl 0579.12012 [21] Parshin, A. N.: On the arithmetic of two-dimensional schemes, Math. USSR Izv. 10(4) (1976), 695-729. · Zbl 0366.14003 [22] Parshin, A. N.: Chem classes, adeles and L-functions, J. Reine Angew. Math. 341 (1983), 174-192. · Zbl 0518.14013 [23] Artin, M., Grothendieck, A. and Verdier, J.-L.: Séminaire de Géométrie Algébrique du Bois-Marie, SGA 4, tome 3, Lecture Notes in Math. 605, Springer-Verlag, Berlin, 1972. [24] Deligne, P.: Séminaire de Géometrie Algébrique du Bois-Marie, SGA 4 1/2, Lecture Notes in Math. 569, Springer-Verlag, Berlin, 1977. [25] Weil, A.: Sur les fonctions algébriques à corps de constantes finis, C. R. Acad. Sci. Paris 210 (1940), 592-594. · JFM 66.0135.01 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.