Abstract
Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including (1) the existence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle theory; (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product; and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.
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[A-B] Andreotti, A. and Bombieri, E.: —Sugli omeomorfismi della varietà algebriche, Ann. Scuola Norm. Sup. Pisa 23 (1969), 431–450.
[B] Barlet, D: Espace analytique rëduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie, Lecture Notes in Math 482, Springer, New York, 1981.
[B-O] Bloch, S. and Ogus, A.: Gersten's conjecture and the homology of schemes, Ann. Eè cole Norm. Sup. 7 (1974), 181–202.
[F1] Friedlander, E.: Algebraic cocycles, Chow varieties, and Lawson homology, Compositio Math. 77 (1991), 55–93.
[F2] Friedlander, E.: Algebraic cocycles on normal, quasi-projective varieties, Compositio Math. 110 (1998), 127–162
[F3] Friedlander, E.: Boch-Ogus properties for topological cycle theory, Ann. École. Norm. Sup. (To appear).
[F-G] Friedlander, E. and Gabber, O.: Cycles spaces and intersection theory, Topol. Methods Modern Math. (1993), 325–370.
[FL-1] Friedlander, E. and Lawson, H. B.: A theory of algebraic cocycles, Ann. of Math. 136 (1992), 361–428.
[FL-2] Friedlander, E. and Lawson, H. B.: Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), 533–565.
[FL-3] Friedlander, E. and Lawson, H. B.: Moving algebraic cycles of bounded degree, Invent. Math. 132 (1998), 91–119.
[F-M] Friedlander, E. and Mazur, B.: Filtratons on the homology of algebraic varieties, Amer. Math. Soc. 529 (1994).
[F-V] Friedlander, E. and Voevodsky, V.: Bivariant cycle cohomology, In: V. Voevodsky, A. Suslin and E. Friedlander (eds), Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., Princeton Univ. Press.
[Fu] Fulton, W.: Intersection Theory, 1984.
[G] Gillet, H.: Riemann-Roch theorems for higher algebraic K-theory, Adv. Math 40 (1981), 203–289.
[H] Hironaka, H.: Triangulations of algebraic sets, In: Algebraic Geometry (Proc. Sympos. Pure Math. 29, Amer. Math. Soc., Providence, 1975, pp. 165–185.
[L-W] Lundell, A. and Weingram, S.: The Topology of CW Complexes, 1969.
[S-V] Suslin, A. and Voevodsky, V.: Chow sheaves, In: V. Voevodsky, A. Suslin and E. Friedlander (eds), Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., Princeton Univ. Press.
[S] Swan, R.: On seminormality, J. Algebra 67 (1980), 210–229.
[V1] Voevodsky, V.: Homology of schemes, Selecta Math. 2 (1996), 111–153.
[V2] Voevodsky, V.: Triangulated category of motives of a field, In: V. Voevodsky, A. Suslin and E. Friedlander (eds), Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., Princeton Univ. Press.
[W] Weibel, C.: Products in Higher Chow groups and motivic cohomology, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, to appear.
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Friedlander, E.M., Walker, M.E. Function Spaces and Continuous Algebraic Pairings for Varieties. Compositio Mathematica 125, 69–110 (2001). https://doi.org/10.1023/A:1002464407035
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DOI: https://doi.org/10.1023/A:1002464407035