Local families of K3 surfaces and applications

Author:
Keiji Oguiso

Journal:
J. Algebraic Geom. **12** (2003), 405-433

DOI:
https://doi.org/10.1090/S1056-3911-03-00362-X

Published electronically:
February 25, 2003

MathSciNet review:
1966023

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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.

[BPV]BPV W. Barth, C. Peters, A. Van de Ven, *Compact complex surfaces*, Springer-Verlag (1984).
[Be]Be A. Beauville, *Variétés Kählerian dont la premiere class de Chern est nulle*, J. Diff. Geom. **18** (1983) 755-782.
[Bo]Bo F. Bogomolov, *Hamiltonian Kähler manifolds*, Soviet. Math. Dokl. **19** (1978) 1462-1465.
[Br]Br C. Borcea, *Homogeneous vector bundles and families of Calabi-Yau threefolds, II*, Proc. Sym. Pure Math. **52** (1991) 83-91.
[BKPS]BKPS R. E. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron, *Families of K3 surfaces*, J. Alg. Geom. **7** (1998) 183-193.
[BC]BC A. Borel and Harish-Chandra, *Arithmetic subgroups of algebraic groups*, Ann. Math. **75** (1962) 485-535.
[CP]CP F. Campana and T. Peternell, *Algebraicity of the ample cone of projective varieties*, J. reine angew. Math. **407** (1990) 160-166.
[CD]CD F. R. Cossec, I. V. Dolgachev, *Enriques surfaces I*, Birkhäuser (1989).
[Co]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]FG W. Fischer, H. Grauert, *Lokal-triviale Familien komplexer Mannighaltigkeiten (German)*, Nachr. Akad. Wiss. Göttingen, II Math.-Phys. Kl. (1965) 89-94.
[Fu]Fu A. Fujiki, *Finite automorphism groups of complex tori of dimension two*, Publ. RIMS Kyoto Univ. **24** (1988) 1-97.
[Gr]Gr A. Grothendieck, *Fondements de la Géométrie Algébrique*, Sec. Math. Paris (1962).
[HLOY]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]Hu D. Huybrechts, *Compact hyperkähler manifolds: Basic results*, Invent. Math. **135** (1999) 63-113.
[Ke]Ke J. H. Keum, *Automorphisms of Jacobian Kummer surfaces*, Compositio Math. **107** (1997) 269-288.
[KK]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]Ko K. Kodaira, *On the structure of compact complex analytic surfaces, I*, Amer. J. Math. **86** (1964) 751-798.
[Kl]Kl J. Kollár, *Rational curves on algebraic varieties. A series of Modern Surveys in Mathematics*, Springer-Verlag **32** (1996).
[Kn1]Kn1 S. Kondo, *Algebraic K3 surfaces with finite automorphism groups*, Nagoya Math. J. **116** (1989) 1-15.
[Kn2]Kn2 S. Kondo, *Niemeier Lattices, Mathieu groups, and finite groups of symplectic automorphisms of $K3$ surfaces*, Duke Math. J. **92** (1998) 593-598.
[Kn3]Kn3 S. Kondo, *The automorphism groups of a generic Kummer surface*, J. Alg. Geom. **7** (1998) 589-609.
[Kv]Kv S. Kovacs, *The cone of curves of a K3 surface*, Math. Ann. **300** (1994) 681-691.
[Mc]Mc C. T. McMullen, *Dynamics on $K3$ surfaces: Salem numbers and Siegel disks*, J. Reine Angew. Math. **545** (2002) 201-233.
[Mu]Mu S. Mukai, *Finite groups of automorphisms of $K3$ surfaces and the Mathieu group*, Invent. Math. **94** (1988) 183-221.
[Ni1]Ni1 V. V. Nikulin, *Finite automorphism groups of Kähler $K3$ surfaces*, Trans. Moscow Math. Soc. **38** (1980) 71-135.
[Ni2]Ni2 V. V. Nikulin, *Integral symmetric bilinear forms and some of their geometric applications*, Math. USSR Izv. **14** (1980) 103-167.
[Ni3]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]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]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]Ni6 V. V. Nikulin, *A remark on algebraic surfaces with polyhedral Mori cone*, Nagoya Math. J. **157** (2000) 73-92.
[Ns]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]OS K. Oguiso, T. Shioda, *The Mordell-Weil lattice of a rational elliptic surface*, Comment. Math. Univ. St. Paul. **40** (1991) 83-99.
[OV]OV K. Oguiso, E. Viehweg, *On the isotriviality of families of elliptic surfaces*, J. Alg. Geom. **10** (2001) 569-598.
[OZ]OZ K. Oguiso, D. Q. Zhang, *K3 surfaces with order 11 automorphisms*, math.AG/ 9907020.
[PSS]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]Sh1 T. Shioda, *On elliptic modular surfaces*, J. Math. Soc. Japan **24** (1972) 20-59.
[Sh2]Sh2 T. Shioda, *On the Mordell-Weil lattices*, Comment. Math. Univ. St. Paul. **39** (1990) 211-240.
[Sh3]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]SI T. Shioda, H. Inose, *On singular K3 surfaces*: In complex analysis and algebraic geometry, Iwanami Shoten (1977) 119-136.
[SM]SM T. Shioda and N. Mitani, *Singular abelian surfaces and binary quadratic forms*, Lect. Notes Math. **412** (1974) 259-287.
[St]St H. Sterk, *Finiteness results for algebraic K3 surfaces*, Math. Z. **189** (1985) 507-513.
[Ta]Ta T. Takagi, *Algebraic integer theory (second edition, in Japanese)*, Iwanami Shoten (1971).
[Vi]Vi E. B. Vinberg, *The two most algebraic $K3$ surfaces*, Math. Ann. **265** (1983) 1-21.
[Yo]Yo H. Yoshihara, *Structure of complex tori with the automorphisms of maximal degree*, Tsukuba J. Math. **4** (1980) 303-311.

[BPV]BPV W. Barth, C. Peters, A. Van de Ven, *Compact complex surfaces*, Springer-Verlag (1984).
[Be]Be A. Beauville, *Variétés Kählerian dont la premiere class de Chern est nulle*, J. Diff. Geom. **18** (1983) 755-782.
[Bo]Bo F. Bogomolov, *Hamiltonian Kähler manifolds*, Soviet. Math. Dokl. **19** (1978) 1462-1465.
[Br]Br C. Borcea, *Homogeneous vector bundles and families of Calabi-Yau threefolds, II*, Proc. Sym. Pure Math. **52** (1991) 83-91.
[BKPS]BKPS R. E. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron, *Families of K3 surfaces*, J. Alg. Geom. **7** (1998) 183-193.
[BC]BC A. Borel and Harish-Chandra, *Arithmetic subgroups of algebraic groups*, Ann. Math. **75** (1962) 485-535.
[CP]CP F. Campana and T. Peternell, *Algebraicity of the ample cone of projective varieties*, J. reine angew. Math. **407** (1990) 160-166.
[CD]CD F. R. Cossec, I. V. Dolgachev, *Enriques surfaces I*, Birkhäuser (1989).
[Co]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]FG W. Fischer, H. Grauert, *Lokal-triviale Familien komplexer Mannighaltigkeiten (German)*, Nachr. Akad. Wiss. Göttingen, II Math.-Phys. Kl. (1965) 89-94.
[Fu]Fu A. Fujiki, *Finite automorphism groups of complex tori of dimension two*, Publ. RIMS Kyoto Univ. **24** (1988) 1-97.
[Gr]Gr A. Grothendieck, *Fondements de la Géométrie Algébrique*, Sec. Math. Paris (1962).
[HLOY]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]Hu D. Huybrechts, *Compact hyperkähler manifolds: Basic results*, Invent. Math. **135** (1999) 63-113.
[Ke]Ke J. H. Keum, *Automorphisms of Jacobian Kummer surfaces*, Compositio Math. **107** (1997) 269-288.
[KK]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]Ko K. Kodaira, *On the structure of compact complex analytic surfaces, I*, Amer. J. Math. **86** (1964) 751-798.
[Kl]Kl J. Kollár, *Rational curves on algebraic varieties. A series of Modern Surveys in Mathematics*, Springer-Verlag **32** (1996).
[Kn1]Kn1 S. Kondo, *Algebraic K3 surfaces with finite automorphism groups*, Nagoya Math. J. **116** (1989) 1-15.
[Kn2]Kn2 S. Kondo, *Niemeier Lattices, Mathieu groups, and finite groups of symplectic automorphisms of $K3$ surfaces*, Duke Math. J. **92** (1998) 593-598.
[Kn3]Kn3 S. Kondo, *The automorphism groups of a generic Kummer surface*, J. Alg. Geom. **7** (1998) 589-609.
[Kv]Kv S. Kovacs, *The cone of curves of a K3 surface*, Math. Ann. **300** (1994) 681-691.
[Mc]Mc C. T. McMullen, *Dynamics on $K3$ surfaces: Salem numbers and Siegel disks*, J. Reine Angew. Math. **545** (2002) 201-233.
[Mu]Mu S. Mukai, *Finite groups of automorphisms of $K3$ surfaces and the Mathieu group*, Invent. Math. **94** (1988) 183-221.
[Ni1]Ni1 V. V. Nikulin, *Finite automorphism groups of Kähler $K3$ surfaces*, Trans. Moscow Math. Soc. **38** (1980) 71-135.
[Ni2]Ni2 V. V. Nikulin, *Integral symmetric bilinear forms and some of their geometric applications*, Math. USSR Izv. **14** (1980) 103-167.
[Ni3]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]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]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]Ni6 V. V. Nikulin, *A remark on algebraic surfaces with polyhedral Mori cone*, Nagoya Math. J. **157** (2000) 73-92.
[Ns]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]OS K. Oguiso, T. Shioda, *The Mordell-Weil lattice of a rational elliptic surface*, Comment. Math. Univ. St. Paul. **40** (1991) 83-99.
[OV]OV K. Oguiso, E. Viehweg, *On the isotriviality of families of elliptic surfaces*, J. Alg. Geom. **10** (2001) 569-598.
[OZ]OZ K. Oguiso, D. Q. Zhang, *K3 surfaces with order 11 automorphisms*, math.AG/ 9907020.
[PSS]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]Sh1 T. Shioda, *On elliptic modular surfaces*, J. Math. Soc. Japan **24** (1972) 20-59.
[Sh2]Sh2 T. Shioda, *On the Mordell-Weil lattices*, Comment. Math. Univ. St. Paul. **39** (1990) 211-240.
[Sh3]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]SI T. Shioda, H. Inose, *On singular K3 surfaces*: In complex analysis and algebraic geometry, Iwanami Shoten (1977) 119-136.
[SM]SM T. Shioda and N. Mitani, *Singular abelian surfaces and binary quadratic forms*, Lect. Notes Math. **412** (1974) 259-287.
[St]St H. Sterk, *Finiteness results for algebraic K3 surfaces*, Math. Z. **189** (1985) 507-513.
[Ta]Ta T. Takagi, *Algebraic integer theory (second edition, in Japanese)*, Iwanami Shoten (1971).
[Vi]Vi E. B. Vinberg, *The two most algebraic $K3$ surfaces*, Math. Ann. **265** (1983) 1-21.
[Yo]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

Received by editor(s):
November 8, 2000

Published electronically:
February 25, 2003

Dedicated:
Dedicated to Professor Yujiro Kawamata on the occasion of his fiftieth birthday