## The moduli and the global period mapping of surfaces with $$K^2=p_g=1$$. A counterexample to the global Torelli problem.(English)Zbl 0444.14008

### MSC:

 14D22 Fine and coarse moduli spaces 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 32J15 Compact complex surfaces 14J15 Moduli, classification: analytic theory; relations with modular forms 14J25 Special surfaces 14J10 Families, moduli, classification: algebraic theory 14H40 Jacobians, Prym varieties 32G20 Period matrices, variation of Hodge structure; degenerations

Zbl 0423.14019
Full Text:

### References:

 [1] A. Andreotti : On a theorem of Torelli . Am. J. of Math., 80 (1958) 801-828. · Zbl 0084.17304 [2] E. Bombieri : Canonical models of surfaces of general type . Publ. Math. I.H.E.S. 42 (1973) 171-219. · Zbl 0259.14005 [3] F. Catanese : Surfaces with K2 = pg = 1 and their period mapping, in Algebraic Geometry, Proc. Copenhagen 1978 , Springer Lect. Notes in Math. n.732 (1979) 1-26. · Zbl 0423.14019 [4] I. Dolgachev : Weighted projective varieties , (to appear). · Zbl 0516.14014 [5] F. Enriques : Le superficie algebriche di genere lineare p(1) = 2 . Rend. Acc. Lincei, s. 5a, vol. VI (1897) 139-144. · JFM 28.0558.05 [6] F. Enriques : Le superficie algebriche . Zanichelli, Bologna, (1949). · Zbl 0036.37102 [7] D. Gieseker : Global moduli for surfaces of general type . Inv. Math. 43 (1977) 233-282. · Zbl 0389.14006 [8] P. Griffiths : Periods of integrals on algebraic manifolds, I, II . Am. J. of Math. 90 (1968) 568-626, 805-865. · Zbl 0183.25501 [9] P. Griffiths : Periods of integrals on algebraic manifolds: summary of main results and discussion of open problems , Bull. Am. Math. Soc. 76 (1970) 228-296. · Zbl 0214.19802 [10] P. Griffiths and W. Schmid : Recent developments in Hodge theory: a discussion of techniques and results . Proc. Int. Coll. Bombay, (1973), Oxford Univ. Press. · Zbl 0355.14003 [11] E. Horikawa : On the periods of Enriques surfaces, I, II . Math. Ann. vol. 234, 235 (1978) 73-88, 217-246. · Zbl 0412.14015 [12] K. Kodaira : Pluricanonical systems on algebraic surfaces of general type . J. Math. Soc. Japan 20 (1968) 170-192. · Zbl 0157.27704 [13] M. Kuranishi : New proof for the existence of locally complete families of complex structures . Proc. Conf. Compl. Analysis, Minneapolis, pp. 142-154, Springer (1965). · Zbl 0144.21102 [14] V.I. Kynef : An example of a simply connected surface of general type for which the local Torelli theorem does not hold . C.R. Ac. Bulg. Sc. 30, n.3 (1977) 323-325. · Zbl 0363.14005 [15] S. Mori : On a generalization of complete intersections . J. Math. Kyoto Univ. 15, n.3 (1975) 619-646. · Zbl 0332.14019 [16] D. Mumford : The canonical ring of an algebraic surface . Annals of Math. 76 (1962) 612-615. [17] C. Peters : The local Torelli theorem, a review of known results in Variètès analytiques compactes , Nice 1977. Springer Lect. Notes in Math. 683 (1978) 62-73. · Zbl 0399.32017 [18] I.I. Piatetski Shapiro and I.R. Shafarevitch : Theorem of Torelli on algebraic surfaces of type K3 , Math. USSR Izvestija 5 (1971) 547-588. · Zbl 0253.14006 [19] G.N. Tjurina : Resolution of singularities of flat deformations of rational double points . Funk. Anal. i Pril . 4, n.1, pp. 77-83. · Zbl 0221.32008 [20] S. Usui : Local Torelli theorem for some non-singular weighted complete intersections . Proceed. Internat. Symposium Algebraic Geometry, Kyoto, 1977. Ed. M. Nagata. Kinokuniya Book-Store, Tokyo, Japan, 1978: pp. 723-734. · Zbl 0418.14005 [21] J.J. Wavrik : Obstructions to the existence of a space of moduli , Global Analysis. Prin. Math. Series n.29 (1969) 403-414. · Zbl 0191.38003 [22] A. Weil : Zum Beweis des Torellischen Satz . Göttingen Nachrichten (1957) 33-53. · Zbl 0079.37002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.