Families of algebraic varieties and invariant cycles

G. A. Mustafin

Izvestiya. Mathematics , Tome 27 (1986) no. 2, pp. 251-278.

Résumé

This paper studies monodromy-invariant cycles in the cohomology of fibers of a family of algebraic varieties. It is shown that the localization of invariant cycles in a neighborhood of a degeneration of the family is a morphism of Hodge structures. An application of this result is the geometric analogue of the Mumford–Tate conjecture for families with strong degenerations. A large class of nonconstant abelian schemes for which the geometric analogue of the Mumford–Tate conjecture holds is constructed. Bibliography: 29 titles.

Bibliographie

[1] Arakelov S. Yu., “Semeistva algebraicheskikh krivykh s fiksirovannymi vyrozhdeniyami”, Izv. AN SSSR. Ser. matem., 35:6 (1971), 1269–1293 | MR | Zbl

[2] Burbaki N., Gruppy i algebry Li, gl. 7, 8, Mir, M., 1978 | MR

[3] Delin P., “Teoriya Khodzha. II”, Matematika, 17:5 (1973), 3–56 | MR

[4] Dynkin E. B., “Struktura poluprostykh algebr Li”, Uspekhi matem. nauk, 2 (1947), 56–127 | MR

[5] Zarkhin Yu. G., Kogomologii algebraicheskikh mnogoobrazii i predstavleniya poluprostykh algebr Li, Preprint NTsBI AN SSSR, Puschino, 1980 | Zbl

[6] Zarkhin Yu. G., O gruppe Khodzha poverkhnosti tipa $K3$, Preprint NTsBI AN SSSR, Puschino, 1982 | MR

[7] Mamford D., Abelevy mnogoobraziya, Mir, M., 1971

[8] Mamford D., “Ob uravneniyakh, opredelyayuschikh abelevy mnogoobraziya. III”, Matematika, 15:2 (1971), 29–56

[9] Mamford D., “Abstraktnye teta-funktsii”, Matematika, 15:6 (1971), 3–11 | MR | Zbl

[10] Mustafin G. A., “Nearkhimedova uniformizatsiya”, Matem. sb., 105(147):2 (1978), 207–237 | MR

[11] Serr Zh. P., Teit Dzh., “Khoroshaya reduktsiya abelevykh mnogoobrazii”, Matematika, 15:5 (1971), 140–165 | Zbl

[12] Teit Dzh., “Endomorfizmy abelevykh mnogoobrazii nad konechnymi polyami”, Matematika, 12:6 (1968), 31–40

[13] Tankeev S. G., “Algebraicheskie tsikly na poverkhnostyakh i abelevykh mnogoobraziyakh”, Izv. AN SSSR. Ser. matem., 45:2 (1981), 398–434 | MR | Zbl

[14] Borel A., “Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem”, Diff. Geometry, 6 (1972), 543–560 | MR | Zbl

[15] Clemens C. H., “Degeneration of kähler manifolds”, Duke Math. Jour., 44:2 (1977), 215–290 | DOI | MR | Zbl

[16] Deligne P., “Theorie de Hodge. III”, Publ. Math. IHES, 1974, no. 44, 5–77 | MR | Zbl

[17] Deligne P., “Varieties de Shimura: interpretation modulaire at techniques de construction de modeles canonique”, Proc. Symp. Pure Math., 33, 1979, 247–290 | MR | Zbl

[18] Gerritzen L., “On non-archimedean representations of abelian varieties”, Math. Ann., 196 (1972), 323–346 | DOI | MR | Zbl

[19] Gerritzen L., “On multiplication algebras of Riemann matrices”, Math. Ann., 194 (1971), 109–122 | DOI | MR | Zbl

[20] Griffiths P., “Periods of integrals on algebraic manifolds. III”, Publ. Math. IHES, 1970, no. 38, 125–180 | MR | Zbl

[21] Grothendieck A., “Hodge's general conjecture is false for trivial reasons”, Topology, 8:3 (1969), 299–303 | DOI | MR | Zbl

[22] Kempf G., Knudsen F., Mumford D., Saint-Donat B., Toroidal embeddings. I, Lect. Notes Math., 339, 1973 | MR | Zbl

[23] Mumford D., “Families of abelian varieties”, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, 347–351 | MR

[24] Satake I., “Holomorphic imbeddings of symmetric domains into a Siegel spase”, Amer. J. Math., 87 (1965), 425–461 | DOI | MR | Zbl

[25] Shimura G., “Moduli of abelian varieties and number theory”, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, 312–332 | MR

[26] Schmid W., “Variation of Hodge Structure: The singularities of the period mapping”, Inv. Math., 22 (1973), 211–319 | DOI | MR | Zbl

[27] Steenbrink J., “Limits of Hodge Structures”, Inv. Math., 31 (1976), 229–257 | DOI | MR | Zbl