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Topological Properties of Cyclic Coverings Branched Along An Ample Divisor

Published online by Cambridge University Press:  20 November 2018

Antonio Lanteri  and
Daniele C. Struppa
Antonio Lanteri
Affiliation:
Universita di Milano, Milano, Italy
Daniele C. Struppa
Affiliation:
Universita di Milano, Milano, Italy
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Let X’ → X be a finite morphism between two complex connected projective k-folds. Since Π is surjective, the Betti numbers of X and X’ are related as follows

(0.1) bi(X) ≦ bi(X’).

In particular, if Π is a cyclic covering and the branch locus A is an ample divisor, (0.1) is in fact an equality for i ≦ k — 1 (see 1.10 or, more generally, [5] ). It seems natural to look for such coverings satisfying

(0.2) bk(X)= bk(X’).

Let us see what happens for k = 2. In this case (0.2) can be rephrased as

(0.3) 2x(Ox) + h1,1 (X) + g(Δ) = 2,

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 3 , 01 June 1989 , pp. 462 - 479
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Barth, W., Peters, C. and Van, A., de Ven Compact complex surfaces (Springer-Verlag, 1984).Google Scholar
2. Beauville, A., Surfaces algébriques complexes, Astérisque 54 (1978).Google Scholar
3. Beltrametti, M., Lanteri, A. and Palleschi, M., Algebraic surfaces containing an ample divisor of arithmetic genus two, Ark. Mat. 25 (1987).Google Scholar
4. Buium, A., Sur le nombre de Picard des revêtements doubles des surfaces algébriques, C.R. Acad. Sci. Paris 296 (1983), 361–364.Google Scholar
5. Cornalba, M., Un'osservazione sulla topologia dei rivestimenti ciclici di varietà algebriche, Boll. Un. Mat. Ital. 18-A (1981), 323–328.Google Scholar
6. Fujita, T., On topological characterizations of complex projective spaces and affine linear spaces, Proc. Japan Acad. 56 A (1980), 231–234.Google Scholar
7. Hartshorne, R., Algebraic geometry (Springer-Verlag, 1977).Google Scholar
8. Iversen, B., Numerical invariants and multiple planes, Amer. J. Math. 92 (1970), 968–996.Google Scholar
9. Lanteri, A. and Palleschi, M., About the adjunction process for polarized algebraic surfaces, J. Reine Angew. Math. 351 (1984), 15–23.Google Scholar
10. Lanteri, A. and Struppa, D., Projective manifolds with the same homology as Pk , Monatsh. Math. 101 (1986), 53–58.Google Scholar
11. Reider, I., Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. 727(1988), 309–316.Google Scholar
12. Sommese, A.J., On manifolds that cannot be ample divisors, Math. Ann. 221 (1976), 55–72.Google Scholar
13. Sommese, A.J.,On the adjunction theoretic structure of projective varieties, in Complex analysis and algebraic geometry, Proc., Gôttingen (1985), 175–213. Lect. Notes Math. 1194 (Springer-Verlag, 1986).Google Scholar
14. Wilson, P.M.H., On projective manifolds with the same rational cohomology as P4 , Rend. Sem. Mat. Univ. Politec. Torino (1986), Special Issue, 15–23.Google Scholar