Abstract
We introduce the concept of a twisting cochain and a twisted complex associated to a coherent sheaf. For sheaves of submanifolds these twisted complexes are used to construct on cochain level the Grothendieck theory of dual class and Gysin map. These explicit constructions give, for instance, a local formula for dual class of higher codimensional submanifolds. We prove a refined version of the Hirzebruch Riemann Roch using such local formulas. We also prove a theorem on when global analytic intersection classes can be computed from first order geometric data. This theory will be used to prove the Holomorphic Lefschetz formula (in Part II) and the Hirzebruch Riemann Roch for analytic coherent sheaves.
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Altman, A.B., Kleiman, S.: Introduction to Grothendieck duality theory. Lecture Notes in Mathematics 146. Berlin, Heidelberg, New York: Springer 1970
de Rham, G.: Variétés différentiables. Paris: Hermann 1960
Gabrielov, A.M., Gelfand, I.M., Losik, M.V.: Combinatorial computation of characteristics classes. Funct. Anal. Appl.9, 12–28 (1975)
Grothendieck, A.: Théorèmes de dualité pour les faisceaux algébrique cohérents. Séminaire Bourbaki,t. 9, 1956–1957, No. 149
Grothendieck, A.: Local cohomology. Lecture Notes in Mathematics 41. Berlin, Heidelberg, New York: Springer 1967
Grothendieck, A.: La théorie des classes de Chern. Bull. Soc. Math. France86, 137–154 (1958)
Hartshorne, R.: Residues and duality. Lecture Notes in Mathematics 20. Berlin, Heidelberg, New York: Springer 1966
King, J.: Residues and Chern classes. Proc. of Symp. in Pure Math.27, 91–97 (1975)
Malgrange, B.: Systèmes différentiels à coefficients constants. Séminaire Bourbaki, t. 15, 1962–1963, no. 246
Ramis, J.P., Ruget, G.: Complexe dualisant et théorèmes de dualité en géométrie analytique complexe. Publ. I.H.E.S.38, 77–91 (1970)
Serre, J.P.: Un théorème de dualité. Comm. Math. Helv.29, 9–26 (1955)
Toledo, D., Tong, Y.L.L.: A parametrix for\(\bar \partial\) and Riemann Roch in Čech theory. Topology,15, 273–301 (1976)
Toledo, D., Tong, Y.L.L.: The holomorphic Lefschetz formula. Bull. A.M.S.81, 1133–1135 (1975)
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The first author is supported in part by NSF grants GP-36418X1 and MCS 76-08478. The second by MCS 75-07986 and Sonderforschungsbereich “Theoretische Mathematik” at Bonn University
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Toledo, D., Tong, Y.L.L. Duality and intersection theory in complex manifolds. I. Math. Ann. 237, 41–77 (1978). https://doi.org/10.1007/BF01351557
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DOI: https://doi.org/10.1007/BF01351557