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Local families of K3 surfaces and applications

Author(s): Keiji Oguiso
Journal: J. Algebraic Geom. 12 (2003), 405-433.
Posted: February 25, 2003
Retrieve article in: PDF DVI PostScript

Abstract | References | Additional information

Abstract: We show the density of the jumping loci of the Picard number of the hyperkähler manifold under a small one-dimensional deformation. We then apply this to study certain hierarchy of the Mordell-Weil lattices of Jacobian K3 surfaces and the automorphism groups in a family of K3 surfaces.

References:

[BPV]
W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, Springer-Verlag (1984).

[Be]
A. Beauville, Variétés Kählerian dont la premiere class de Chern est nulle, J. Diff. Geom. 18 (1983) 755-782.

[Bo]
F. Bogomolov, Hamiltonian Kähler manifolds, Soviet. Math. Dokl. 19 (1978) 1462-1465.

[Br]
C. Borcea, Homogeneous vector bundles and families of Calabi-Yau threefolds, II, Proc. Sym. Pure Math. 52 (1991) 83-91.

[BKPS]
R. E. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron, Families of K3 surfaces, J. Alg. Geom. 7 (1998) 183-193.

[BC]
A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math. 75 (1962) 485-535.

[CP]
F. Campana and T. Peternell, Algebraicity of the ample cone of projective varieties, J. reine angew. Math. 407 (1990) 160-166.

[CD]
F. R. Cossec, I. V. Dolgachev, Enriques surfaces I, Birkhäuser (1989).

[Co]
D. A. Cox, Mordell-Weil groups of elliptic curves over $\mathbf{C}(t)$ with $p_{g} = 0$ or $1$, Duke Math. J. 49 (1982) 677-689.

[FG]
W. Fischer, H. Grauert, Lokal-triviale Familien komplexer Mannighaltigkeiten (German), Nachr. Akad. Wiss. Göttingen, II Math.-Phys. Kl. (1965) 89-94.

[Fu]
A. Fujiki, Finite automorphism groups of complex tori of dimension two, Publ. RIMS Kyoto Univ. 24 (1988) 1-97.

[Gr]
A. Grothendieck, Fondements de la Géométrie Algébrique, Sec. Math. Paris (1962).

[HLOY]
S. Hosono, B. H. Lian, K. Oguiso, S. T. Yau, Kummer structures on a K3 surfaces - An old question of T. Shioda, math.AG/0202082.

[Hu]
D. Huybrechts, Compact hyperkähler manifolds: Basic results, Invent. Math. 135 (1999) 63-113.

[Ke]
J. H. Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math. 107 (1997) 269-288.

[KK]
J. H. Keum and S. Kondo, The automorphism groups of Kummer surfaces associated with the product of two elliptic curves, Trans. Amer. Math. Soc. 353 (2001) 1469-1487.

[Ko]
K. Kodaira, On the structure of compact complex analytic surfaces, I, Amer. J. Math. 86 (1964) 751-798.

[Kl]
J. Kollár, Rational curves on algebraic varieties. A series of Modern Surveys in Mathematics, Springer-Verlag 32 (1996).

[Kn1]
S. Kondo, Algebraic K3 surfaces with finite automorphism groups, Nagoya Math. J. 116 (1989) 1-15.

[Kn2]
S. Kondo, Niemeier Lattices, Mathieu groups, and finite groups of symplectic automorphisms of $K3$ surfaces, Duke Math. J. 92 (1998) 593-598.

[Kn3]
S. Kondo, The automorphism groups of a generic Kummer surface, J. Alg. Geom. 7 (1998) 589-609.

[Kv]
S. Kovacs, The cone of curves of a K3 surface, Math. Ann. 300 (1994) 681-691.

[Mc]
C. T. McMullen, Dynamics on $K3$ surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math. 545 (2002) 201-233.
[Mu]
S. Mukai, Finite groups of automorphisms of $K3$ surfaces and the Mathieu group, Invent. Math. 94 (1988) 183-221.

[Ni1]
V. V. Nikulin, Finite automorphism groups of Kähler $K3$ surfaces, Trans. Moscow Math. Soc. 38 (1980) 71-135.

[Ni2]
V. V. Nikulin, Integral symmetric bilinear forms and some of their geometric applications, Math. USSR Izv. 14 (1980) 103-167.

[Ni3]
V. V. Nikulin, On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by the $2$-reflections, J. Soviet Math. 22 (1983) 1401-1476.

[Ni4]
V. V. Nikulin, Surfaces of type K3 with finite automorphism groups and a Picard group of rank three, Proc. Steklov Institute Math. 3 (1985) 131-155.

[Ni5]
V. V. Nikulin, Discrete reflection groups in Lobachevsky spaces and algebraic surfaces: in Proceedings of the International Congress of Mathematics (Berkeley 1986) Amer. Math. Soc. (1987) 654-671.

[Ni6]
V. V. Nikulin, A remark on algebraic surfaces with polyhedral Mori cone, Nagoya Math. J. 157 (2000) 73-92.

[Ns]
K. Nishiyama, Examples of Jacobian fibrations on some K3 surfaces whose Mordell-Weil lattices have the maximal rank 18, Comment. Math. Univ. St. Paul. 44 (1995) 219-223.

[OS]
K. Oguiso, T. Shioda, The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Paul. 40 (1991) 83-99.

[OV]
K. Oguiso, E. Viehweg, On the isotriviality of families of elliptic surfaces, J. Alg. Geom. 10 (2001) 569-598.

[OZ]
K. Oguiso, D. Q. Zhang, K3 surfaces with order 11 automorphisms, math.AG/ 9907020.

[PSS]
I. Piatetski-Shapiro and I. R. Shafarevich, A Torelli Theorem for algebraic surfaces of type K3, Math. USSR Izv. 5 (1971) 547-587.

[Sh1]
T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972) 20-59.

[Sh2]
T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990) 211-240.

[Sh3]
T. Shioda, Theory of Mordell-Weil lattices: in Proceedings of the International Congress of Mathematicians (Kyoto 1990) Math. Soc. Japan (1991) 473-489.

[SI]
T. Shioda, H. Inose, On singular K3 surfaces: In complex analysis and algebraic geometry, Iwanami Shoten (1977) 119-136.

[SM]
T. Shioda and N. Mitani, Singular abelian surfaces and binary quadratic forms, Lect. Notes Math. 412 (1974) 259-287.

[St]
H. Sterk, Finiteness results for algebraic K3 surfaces, Math. Z. 189 (1985) 507-513.

[Ta]
T. Takagi, Algebraic integer theory (second edition, in Japanese), Iwanami Shoten (1971).

[Vi]
E. B. Vinberg, The two most algebraic $K3$ surfaces, Math. Ann. 265 (1983) 1-21.

[Yo]
H. Yoshihara, Structure of complex tori with the automorphisms of maximal degree, Tsukuba J. Math. 4 (1980) 303-311.

Additional Information:

Keiji Oguiso
Affiliation: Department of Mathematical Sciences, University of Tokyo, 153-8914 Komaba Meguro, Tokyo, Japan
Email: oguiso@ms.u-tokyo.ac.jp

PII: S 1056-3911(03)00362-X
Received by editor(s): November 8, 2000
Posted: February 25, 2003
Dedicated: Dedicated to Professor Yujiro Kawamata on the occasion of his fiftieth birthday

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