Download Real Algebraic Geometry: Proceedings of the Conference held by Manfred Knebusch (auth.), Michel Coste, Louis Mahé, PDF

By Manfred Knebusch (auth.), Michel Coste, Louis Mahé, Marie-Françoise Roy (eds.)

Ten years after the 1st Rennes overseas assembly on actual algebraic geometry, the second checked out the advancements within the topic throughout the intervening decade - see the 6 survey papers indexed lower than. additional contributions from the members on contemporary learn coated genuine algebra and geometry, topology of genuine algebraic forms and 16thHilbert challenge, classical algebraic geometry, ideas in genuine algebraic geometry, algorithms in actual algebraic geometry, semialgebraic geometry, genuine analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology within the final ten years.- R. Parimala: Algebraic and topological invariants of actual algebraic varieties.- Polotovskii, G.M.: at the category of decomposing aircraft algebraic curves.- Scheiderer, C.: genuine algebra and its functions to geometry within the final ten years: a few significant advancements and results.- Shustin, E.L.: Topology of actual aircraft algebraic curves.- Silhol, R.: Moduli difficulties in actual algebraic geometry. additional contributions through: S. Akbulut and H. King; C. Andradas and J. Ruiz; A. Borobia; L. Br|cker; G.W. Brumfield; A. Castilla; Z. Charzynski and P. Skibinski; M. Coste and M. Reguiat; A. Degtyarev; Z. Denkowska; J.-P. Francoise and F. Ronga; J.M. Gamboa and C. Ueno; D. Gondard- Cozette; I.V. Itenberg; P. Jaworski; A. Korchagin; T. Krasinksi and S. Spodzieja; ok. Kurdyka; H. Lombardi; M. Marshall and L. Walter; V.F. Mazurovskii; G. Mikhalkin; T. Mostowski and E. Rannou; E.I. Shustin; N. Vorobjov.

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Sithol ([si] Th. , surfaces for which X ~ is birational to iP~), TI(X) = 0 . 40 2 Image of the cycle map; C o n n e c t i o n s w i t h K-Theory Let X be an atCfine algebraic variety over ~ with X ( ~ ) compact. Let KF(X) denote the Grothendieck group of topological F-vector bundles on X ( ~ ) , F being ~ , ¢ or ~-/. By a theorem of Swan ([Sw]l p. 268), KF(X) ~Ko(F®C(X)), the Grothendieck group on finitely generated projective F® C(X)-modules, C(X) denoting the ring of continuous functions on X ( ~ ) .

1) exhibit a large family of real varieties X for which the group Kf-alg(X) is rather small. They show that if X '--* ~ P ~ is a real variety of dimension at least 1 such that X ( ~ ) is connected and X ¢ is a non-singular complete intersection, KF_,Ig(X) is either finite or a direct sum of a finite group and 2~. If dim X is odd and H*'*~(X, 2~) is torsion free, they show that K¢_~lg(X ) = O. However, the groups KF(X) may generally have arbitrarily large rank (cf. [So-B-K]). 3 C e r t a i n t y p e s of algebraic m o d e l s for compact C°%manifolds After the solution of the Nash conjecture by Tognoli [T], the following natural questions, related to approximations of algebraic models by non-singular algebraic sets, were raised (cf.

Math. 191-210 (1982). [MI2 L. Mah6, Th6or&me de Pfister pour les vari6t6s et anneaux de Witt r6duits, Invent. Math. 85, 53-72 (1986). Ann. 260, [O-P-S] M. Ojanguren, R. Parimala and R. Sridharan, Symplectic bundles over affine surfaces, Comment. Math. Helv. 61,491-500 (1986). [PI W. Pardon, A relation between Witt groups and zero cycles in a regular ring, Algebraic K-theory, Geometry and Analysis, SLN 1046, 1984, pp. 261-328. 51 R. Parimala, Witt groups of affine threefolds, Duke Math. J. 57, 947-954 (1988).

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