# On the Kodaira dimension of the moduli space of K3 surfaces

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On the Kodaira dimension of the moduli space of K3 surfaces COMPOSITIO MATHEMATICA SHIGEYUKIKONDO¯ On theKodaira dimension of themoduli space of K3 surfaces Compositio Mathematica, tome 89, no 3 (1993), p. 251-299. <http://www.numdam.org/item?id=CM_1993__89_3_251_0> © Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 251 On the Kodaira dimension of the moduli space of K3 surfaces SHIGEYUKI KONDO Department of Mathematics, Saitama University, Shimo-Okubo 255, Urawa, Saitama 338, Japan Received 16 June 1992; accepted in final form 13 September 1992 Compositio Mathematica 89: 251-299, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands. 0. Introduction A compact complex smooth surface X is called a K3 surface if the canonical line bundle KX is trivial and dim H1(X, (9x) = 0. The period space of algebraic K3 surfaces with a primitive polarization of degree 2d is of the form Yt 2d = !?fi2d/r 2d where !?fi2d is a 19-dimensional bounded symmetric domain of type IV and r 2d is an arithmetic subgroup acting properly discontinuously on -q2d’ The Yt2d is an irreducible normal quasi-projective variety (Baily, Borel [4]). It follows from the Torelli theorem for K3 surfaces (Piatetskii-Shapiro, Shafarevich [18]) and the surjectivity of the period map (Kulikov [10]) that 4’2d is the coarse moduli space of K3 surfaces with a primitive polarization of degree 2d. It is known that ’ ’C 2d is unirational for 1 d 9 or d = 11 (Mukai [11]). The purpose of th

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