Projective regular models for abelian varieties, semistable reduction, and the height pairing

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Projective regular models for abelian varieties, semistable reduction, and the height pairing

Klaus Künnemann

Source: Duke Math. J. Volume 95, Number 1 (1998), 161-212.

First Page: Show

Primary Subjects: 14G25

Secondary Subjects: 11G10, 14C25, 14K15

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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077229507
Mathematical Reviews number (MathSciNet): MR1646554
Zentralblatt MATH identifier: 0955.14017
Digital Object Identifier: doi:10.1215/S0012-7094-98-09505-9

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References

[AN] V. Alexeev and I. Nakamura, On Mumford's construction of degenerating abelian varieties, preprint, 1996.

[Be1]¹ A. A. Beilinson, Height pairing between algebraic cycles, $K$-Theory, Arithmetic and Geometry (Moscow, 1984–86), Lecture Notes in Math., vol. 1289, Springer-Verlag, Berlin, 1987, pp. 1–25.

Mathematical Reviews (MathSciNet): MR89h:11027

Zentralblatt MATH: 0651.14002

[Be1]² A. A. Beilinson, Height pairing between algebraic cycles, Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., Providence, 1987, pp. 1–24.

Mathematical Reviews (MathSciNet): MR89g:11052

Zentralblatt MATH: 0624.14005

[Be2] A. A. Beilinson, Higher regulators and values of $L$-functions, J. Soviet Math. 30 (1985), 2036–2070.

Mathematical Reviews (MathSciNet): MR760999

[Blo] S. Bloch, Height pairings for algebraic cycles, J. Pure Appl. Algebra 34 (1984), no. 2-3, 119–145.

Mathematical Reviews (MathSciNet): MR86h:14015

Zentralblatt MATH: 0577.14004

Digital Object Identifier: doi:10.1016/0022-4049(84)90032-X

[BGS1] S. Bloch, H. Gillet, and C. Soulé, Nonarchimedean Arakelov theory, J. Algebraic Geom. 4 (1995), no. 3, 427–485.

Mathematical Reviews (MathSciNet): MR96g:14019

Zentralblatt MATH: 0866.14011

[BGS2] S. Bloch, H. Gillet, and C. Soulé, Algebraic cycles on degenerate fibers, Arithmetic Geometry (Cortona, 1994) ed. F. Catanese, Sympos. Math., vol. 37, Cambridge Univ. Press, Cambridge, 1997, pp. 45–69.

Mathematical Reviews (MathSciNet): MR98i:14011

Zentralblatt MATH: 0955.14007

[BLR] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Ergeb. Math. Grenzgeb. (3), vol. 21, Springer-Verlag, Berlin, 1990.

Mathematical Reviews (MathSciNet): MR91i:14034

Zentralblatt MATH: 0705.14001

[Br] L. Breen, Fonctions thêta et théorème du cube, Lecture Notes in Math., vol. 980, Springer-Verlag, Berlin, 1983.

Mathematical Reviews (MathSciNet): MR87d:14036

Zentralblatt MATH: 0558.14029

[B1] J.-L. Brylinski, Eventails et variétés toriques, Séminaire sur les singularités des surfaces (Palaiseau, 1976/77), Lecture Notes in Math., vol. 777, Springer-Verlag, Berlin, 1980, pp. 247–288.

Zentralblatt MATH: 0431.14013

[B2] J.-L. Brylinski, Décomposition simpliciale d'un réseau, invariante par un groupe fini d'automorphismes, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 2, A137–A139.

Mathematical Reviews (MathSciNet): MR80f:14024

Zentralblatt MATH: 0406.14022

[Ch] C. L. Chai, Compactification of Siegel Moduli Schemes, London Math. Soc. Lecture Note Ser., vol. 107, Cambridge Univ. Press, Cambridge, 1985.

Mathematical Reviews (MathSciNet): MR88b:32074

Zentralblatt MATH: 0578.14009

[De]¹ P. Deligne, Théorie de Hodge, III, Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5–77.

Mathematical Reviews (MathSciNet): MR58:16653b

Zentralblatt MATH: 0237.14003

Digital Object Identifier: doi:10.1007/BF02685881

[De]² P. Deligne, La conjecture de Weil, II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252.

Mathematical Reviews (MathSciNet): MR83c:14017

Zentralblatt MATH: 0456.14014

Digital Object Identifier: doi:10.1007/BF02684780

[FC] G. Faltings and C. L. Chai, Degeneration of abelian varieties, Ergeb. Math. Grenzgeb (3), vol. 22, Springer-Verlag, Berlin, 1990.

Mathematical Reviews (MathSciNet): MR92d:14036

Zentralblatt MATH: 0744.14031

[Fu] W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3), vol. 2, Springer-Verlag, Berlin, 1984.

Mathematical Reviews (MathSciNet): MR85k:14004

Zentralblatt MATH: 0541.14005

[FL] W. Fulton and S. Lang, Riemann-Roch Algebra, Grundlehren Math. Wiss., vol. 277, Springer-Verlag, New York, 1985.

Mathematical Reviews (MathSciNet): MR88h:14011

Zentralblatt MATH: 0579.14011

[FMSS] W. Fulton, R. MacPherson, F. Sottile, and B. Sturmfels, Intersection theory on spherical varieties, J. Algebraic Geom. 4 (1995), no. 1, 181–193.

Mathematical Reviews (MathSciNet): MR95j:14004

Zentralblatt MATH: 0819.14019

[GS1] H. Gillet and C. Soulé, Intersection theory using Adams operations, Invent. Math. 90 (1987), no. 2, 243–277.

Mathematical Reviews (MathSciNet): MR89d:14005

Zentralblatt MATH: 0632.14009

Digital Object Identifier: doi:10.1007/BF01388705

[GS2] H. Gillet and C. Soulé, Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 93–174.

Mathematical Reviews (MathSciNet): MR92d:14016

Zentralblatt MATH: 0741.14012

[Gi] J. Giraud, Cohomologie non abélienne, Grundlehren Math. Wiss., vol. 179, Springer-Verlag, Berlin, 1971.

Mathematical Reviews (MathSciNet): MR49:8992

Zentralblatt MATH: 0226.14011

[Gr] W. Green, Heights in families of abelian varieties, Duke Math. J. 58 (1989), no. 3, 617–632.

Mathematical Reviews (MathSciNet): MR90k:14044

Zentralblatt MATH: 0698.14043

Digital Object Identifier: doi:10.1215/S0012-7094-89-05829-8

Project Euclid: euclid.dmj/1077307671

[GrS] B. H. Gross and C. Schoen, The modified diagonal cycle on the triple product of a pointed curve, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 3, 649–679.

Mathematical Reviews (MathSciNet): MR96e:14008

Zentralblatt MATH: 0822.14015

[G] A. Grothendieck, Standard conjectures on algebraic cycles, Algebraic Geometry (Bombay, 1968) ed. S. S. Abhyankhar, et al., Oxford Univ. Press, Oxford, 1969, pp. 193–199.

Mathematical Reviews (MathSciNet): MR42:3088

Zentralblatt MATH: 0201.23301

[G2] A. Grothendieck, Groupes de monodromie en géometrie algébrique, I, Lecture Notes in Math., vol. 288, Springer-Verlag, Berlin, 1972.

Mathematical Reviews (MathSciNet): MR50:7134

Zentralblatt MATH: 0237.00013

[GAV]¹ A. Grothendieck, M. Artin, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Math., vol. 269, Springer-Verlag, Berlin, 1972.

Mathematical Reviews (MathSciNet): MR50:7130

Zentralblatt MATH: 0234.00007

[GAV]² A. Grothendieck, M. Artin, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Math., vol. 270, Springer-Verlag, Berlin, 1972.

Mathematical Reviews (MathSciNet): MR50:7131

Zentralblatt MATH: 0237.00012

[GAV]³ A. Grothendieck, M. Artin, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Math., vol. 305, Springer-Verlag, Berlin, 1973.

Mathematical Reviews (MathSciNet): MR50:7132

Zentralblatt MATH: 0245.00002

[GD]¹ A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. (1960), no. 4, 228.

Mathematical Reviews (MathSciNet): MR36:177a

Zentralblatt MATH: 0118.36206

[GD]² A. Grothendieck and J. Dieudonné, Éléments de géométrie algebrique, II, Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222.

Mathematical Reviews (MathSciNet): MR29:1208

Zentralblatt MATH: 0118.36206

[GD]³ A. Grothendieck and J. Dieudonné, Éléments de géométrie algebrique, III, Étude cohomologique des faisceaux cohérents, I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167.

Mathematical Reviews (MathSciNet): MR29:1209

Zentralblatt MATH: 0118.36206

[GD]⁴ A. Grothendieck and J. Dieudonné, Éléments de géométrie algebrique, III, Étude cohomologique des faisceaux cohérents, II, Inst. Hautes Études Sci. Publ. Math. 17 (1963), 91.

Mathematical Reviews (MathSciNet): MR29:1210

Zentralblatt MATH: 0122.16102

[GD]⁵ A. Grothendieck and J. Dieudonné, Éléments de géométrie algebrique, IV, Étude locale des schémas et des morphismes de schémas, II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231.

Mathematical Reviews (MathSciNet): MR33:7330

Zentralblatt MATH: 0135.39701

[GD]⁶ A. Grothendieck and J. Dieudonné, Éléments de géométrie algebrique, IV, Étude locale des schémas et des morphismes de schémas, III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255.

Mathematical Reviews (MathSciNet): MR36:178

Zentralblatt MATH: 0144.19904

[GD]⁷ A. Grothendieck and J. Dieudonné, Éléments de géométrie algebrique, IV, Étude locale des schémas et des morphismes de schémas, IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361.

Mathematical Reviews (MathSciNet): MR39:220

Zentralblatt MATH: 0153.22301

[H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, Berlin, 1971.

Mathematical Reviews (MathSciNet): MR463157

[dJ] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93.

Mathematical Reviews (MathSciNet): MR98e:14011

Zentralblatt MATH: 0916.14005

Digital Object Identifier: doi:10.1007/BF02698644

[JM] B. W. Jordan and D. R. Morrison, On the Néron models of abelian surfaces with quaternionic multiplication, J. Reine Angew. Math. 447 (1994), 1–22.

Mathematical Reviews (MathSciNet): MR95f:14086

Zentralblatt MATH: 0791.14021

[KKMS] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal Embeddings, I, Lecture Notes in Math., vol. 339, Springer-Verlag, Berlin, 1973.

Mathematical Reviews (MathSciNet): MR49:299

Zentralblatt MATH: 0271.14017

[KT] S. L. Kleiman and A. Thorup, Intersection theory and enumerative geometry: A decade in review, Algebraic Geometry (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, 1987, pp. 321–370.

Mathematical Reviews (MathSciNet): MR90f:14034

Zentralblatt MATH: 0664.14031

[K1] K. Künnemann, Higher Picard varieties and the height pairing, Amer. J. Math. 118 (1996), no. 4, 781–797.

Mathematical Reviews (MathSciNet): MR97d:14064

Zentralblatt MATH: 0876.14006

Digital Object Identifier: doi:10.1353/ajm.1996.0033

[K2] K. Künnemann, The Kähler identity for bigraded Hodge-Lefschetz modules and its application in nonarchimedean Arakelov geometry, to appear in J. Algebraic Geom.

[L] G. Laumon, Homologie étale, Séminaire de geometrie analytique (École Norm. Sup., Paris, 1974–1975), Astérisque, vol. 36–37, Soc. Math. France, Paris, 1976, pp. 163–188.

Mathematical Reviews (MathSciNet): MR56:3017

Zentralblatt MATH: 0341.14006

[Ma] H. Matsumura, Commutative Ring Theory, Cambridge Stud. Adv. Math., vol. 8, Cambridge Univ. Press, Cambridge, 1986.

Mathematical Reviews (MathSciNet): MR88h:13001

Zentralblatt MATH: 0603.13001

[MB] L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129 (1985), 266, Soc. Math. France, Paris.

Mathematical Reviews (MathSciNet): MR87j:14069

Zentralblatt MATH: 0595.14032

[M1] D. Mumford, Abelian Varieties, Tata Inst. Fund. Res. Stud. Math., vol. 5, Oxford Univ. Press, London, 1970.

Mathematical Reviews (MathSciNet): MR44:219

Zentralblatt MATH: 0223.14022

[M2] D. Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239–272.

Mathematical Reviews (MathSciNet): MR50:4593

Zentralblatt MATH: 0241.14020

[Ra] M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Mathematics, Vol. 119, Springer-Verlag, Berlin, 1970.

Mathematical Reviews (MathSciNet): MR41:5381

Zentralblatt MATH: 0195.22701

[Ta] J. Tate, Conjectures on algebraic cycles in $l$-adic cohomology, Motives (Seattle, Wash., 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, 1994, pp. 71–83.

Mathematical Reviews (MathSciNet): MR95a:14010

Zentralblatt MATH: 0814.14009

[T] T. Tsuji, $p$-adic étale cohomology and crystalline cohomology in the semistable reduction case, preprint, 1996.

[V] J.-L. Verdier, Le théorème de Riemann-Roch pour les intersections complètes, Séminaire de geometrie analytique (École Norm. Sup., Paris, 1974–1975), Astérisque, vol. 36-37, Soc. Math. France, Paris, 1976, pp. 189–228.

Mathematical Reviews (MathSciNet): MR56:3007

Zentralblatt MATH: 0334.14026

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