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An example of two homeomorphic, nondiffeomorphic complete intersection surfaces. (English) Zbl 0707.14045
Let \(d_ i\in {\mathbb{N}}\), \(d_ i\geq 2\), \(i=1,...,r\). A complete intersection surface of multidegree \((d_ 1,...,d_ r)\) is a nonsingular algebraic surface \(S(d_ 1,...,d_ r)\subset {\mathbb{C}}{\mathbb{P}}^{r+2}\) which is the transversal intersection of r hypersurfaces of degree \(d_ 1,...,d_ r,\) respectively. Any two complete intersection surfaces with the same multidegree are diffeomorphic. The author proves (by using results of Freedman, Donaldson, Friedman and Morgan) that \(S(10,7,7,6,3,3)\subset {\mathbb{C}}{\mathbb{P}}^ 8\) and \(S(9,5,3,3,3,3,3,2,2)\subset {\mathbb{C}}{\mathbb{P}}^{11}\) are homeomorphic, but not diffeomorphic.
Reviewer: T.Krasiński

MSC:
14M10 Complete intersections
57R50 Differential topological aspects of diffeomorphisms
14J10 Families, moduli, classification: algebraic theory
14N05 Projective techniques in algebraic geometry
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References:
[1] Barth, W., Peters, C., Van de Ven, A.: compact complex surfaces. Berlin Heidelberg New York Tokyo: Springer 1984
[2] Beauville, A.: Le groupe de monodromie des familles universelles d’hypersurfaces et d’intersections complètes. In: Grauert, H. (ed.) complex analysis and algebraic geometry. Proc. Conf., Göttingen 1985 (Lect. Notes Math., vol1194, pp. 8–18). Berlin Heidelberg New York: Springer 1986
[3] Donaldson, S.K.: Polynomial invariants for smooth four-manifolds. Topology (to appear) · Zbl 0715.57007
[4] Ebeling, W.: An arithmetic characterisation of the symmetric monodromy groups of singularities. Invent. Math.77, 85–99 (1984) · Zbl 0542.14026 · doi:10.1007/BF01389136
[5] Ebeling, W.: Vanishing lattices and monodromy groups of isolated complete intersection singularities. Invent. Math.90, 653–668 (1987) · Zbl 0633.32014 · doi:10.1007/BF01389184
[6] Freedman, M.: The topology of four-dimensional manifolds. J. Differ. Geom.17, 357–454 (1982) · Zbl 0528.57011
[7] Friedman, R., Moishezon, B., Morgan, J.W.: On theC invariance of the canonical classes of certain algebraic surfaces. Bull. Am. Math. Soc. (N.S.)17, 283–286 (1987) · Zbl 0627.57014 · doi:10.1090/S0273-0979-1987-15561-3
[8] Libgober, A.S., Wood, J.W.: Differentiable structures on complete intersections I. Topology21, 469–482 (1982) · Zbl 0504.57015 · doi:10.1016/0040-9383(82)90024-6
[9] Milnor, J.: On simply connected 4-manifolds. Proc. Internat. Sympos. Algebraic Topology, Mexico 1958, pp. 122–128. · Zbl 0105.17204
[10] Moishezon, B.: Analogs of Lefschetz theorems for linear systems with isolated singularities. J. Differ. Geom. (to appear) · Zbl 0704.57009
[11] Nijenhuis, A., Wilf, H.: Combinatorial algorithms. 2nd Edition. New York: Academic Press 1978 · Zbl 0476.68047
[12] Serre, J.-P.: Cours d’arithmétique. Paris: Presses Universitaires de France 1970
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