Cohomology of a~quasihomogeneous complete intersection

A. G. Aleksandrov

Izvestiya. Mathematics , Tome 26 (1986) no. 3, pp. 437-477.

Résumé

In this paper are computed the Poincaré series of the highest local cohomology of the modules of regular forms on a nondegenerate quasihomogeneous singularity and the dimensions of the cohomology spaces of the bidual sheaves of holomorphic forms on a quasismooth complete intersection. Theorems are proved about the structure of the module of vector fields on a nondegenerate quasihomogeneous singularity and about whether the maximal modular stratum of a versal deformation of such a singularity is reduced. Bibliography: 47 titles.

Bibliographie

[1] Altman A., Kleiman S., Introduction to Grothendieck Duality Theory, Lecture Notes in Math., No 146, Springer-Verlag, 1971 | MR

[2] Arnold V. I., “Zamechaniya o metode statsionarnoi fazy i chislakh Kokstera”, Uspekhi matem. nauk, 28:5 (1973), 17–44 | MR

[3] Arnold V. I., “Kriticheskie tochki gladkikh funktsii i ikh normalnye formy”, Uspekhi matem. nauk, 30:5 (1975), 5–65 | MR

[4] Arnold V., “On some problems in singularity theory”, Geometry and Analysis: papers dedicated to the memory of V. K. Patodi, Bombay, 1981, 1–9 | MR

[5] Burbaki H., Kommutativnaya algebra, Mir, M., 1971 | MR

[6] Brückmann P., “Zür Kohomologie von vollständigen Durchschnitten mit Koeffizienten in der Garbe der Keime der Differentialformen. II”, Math. Nachr., 77 (1977), 307–318 | DOI | MR | Zbl

[7] Buchsbaum D. A., Rim D. S., “A generalized Koszul Complex. II. Depth and Multiplicity”, Transactions Amer. Math. Soc., 111:2 (1964), 197–224 | DOI | MR | Zbl

[8] Cartan H., “Quotient d'un espace analytique par un group d'automorphismes”, Algebraic geometry and topology, Princeton University Press, Princeton, N. J., 1957, 90–102 | MR | Zbl

[9] Dolgachev I., “$\mathbf C^*$-surface singularities whose $K$ is a homological sphere”, Notice of Amer. Math. Soc., 25:2 (1978), A–303

[10] Dolgachev I., “Weighted projective varieties”, Group Actions and Vector Fields, Lecture Notes in Math., No 956, Springer-Verlag, 1982 | MR

[11] Flenner H., “Rationale quasihomogene Singularitäten”, Arch. Math., 36 (1981), 35–44 | DOI | MR | Zbl

[12] Flenner H., “Divisorenklassengruppen quasihomogener Singularitäten”, Jour. für die reine und Angew. Math., 328 (1981), 128–160 | MR | Zbl

[13] Goto S., Watanabe K., “On graded rings. I”, Jour. Math. Soc. Japan, 30 (1978), 179–213 | DOI | MR | Zbl

[14] Greuel G.-M., “Der Gauss–Manin–Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten”, Math. Ann., 214:2 (1975), 235–266 | DOI | MR | Zbl

[15] Greuel G.-M., Hamm H. A., “Invarianten quasihomogener vollständiger Durchschnitte”, Invent. Math., 49:1 (1978), 67–86 | DOI | MR | Zbl

[16] Greuel G.-M., “Dualität in der lokalen Kohomologie isolierter Singularitäten”, Math. Ann., 250 (1980), 157–173 | MR | Zbl

[17] Grothendieck A., Local Cohomology, Lecture Notes in Math., No 41, Springer-Verlag, 1967 | MR | Zbl

[18] Khamm G. A., “Rod $\chi_y$ kvaziodnorodnykh polnykh peresechenii”, Funkts. analiz i ego prilozheniya, 11:1 (1977), 87–88 | Zbl

[19] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981 | MR | Zbl

[20] Hirzebruch F., “Der Satz von Riemann–Roch in Faisceau-theoretischer Formulirung”, Proceedings of the International Congress of Mathematicians, v. III (1954, Amsterdam), North Holland Publ. Co., 1956, 457–473 | MR

[21] Kantor J. M., “Dérivations sur les singularités quasi-homogènes: cas des courbes”, C. R. Acad. Sci. Paris Sér. A, 287:16 (1978), 1117–1119 | MR | Zbl

[22] Kantor J. M., “Dérivations sur les singularités quasi-homogènes: cas des hypersurfaces”, C. R. Acad. Sci. Paris Sér. A, 288 (1979), 33–34 | MR | Zbl

[23] Kunz E., “Residuen von Differentialformen auf Cohen–Macoley Varitäten”, Math. Zeit., 152 (1977), 165–189 | DOI | MR | Zbl

[24] Lebelt K., “Freie Auflösungen äusserer Potenzen”, Manuscr. Math., 21 (1977), 341–355 | DOI | MR | Zbl

[25] Naruki I., “Some remarks on isolated Singularity and their Application to algebraic Manifolds”, Publ. RIMS, Kyoto Univ., 13 (1977), 41–70 | DOI | MR

[26] Palamodov V. P., “Moduli i versalnye deformatsii kompleksnykh prostranstv”, Dokl. AN SSSR, 230:1 (1976), 34–37 | MR | Zbl

[27] Palamodov V. P., “Moduli in versal deformation of complex spaces”, Varieties Analytiques Compactes, Lecture Notes in Math., No 683, Springer-Verlag, 1979 | MR

[28] Palamodov V. P., “Kogomologiya analiticheskikh algebr”, Tr. Mosk. matem. o-va, 44 (1982), 3–61 | MR | Zbl

[29] Reid M., “Canonical 3-folds”, Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980, 273–310 | MR

[30] Saito K., “Quasihomogene isolierte Singularitäten von Hyperflächen”, Invent. Math., 14 (1971), 123–142 | DOI | MR | Zbl

[31] Scheja G., Wiebe H., “Über Derivationen von lokalen Analytischen Algebren”, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), XI, Academic Press, London, 1973, 161–192 | MR

[32] Scheja G., Wiebe H., “Über Derivationen in isolierter Singularitäten auf vollständigen Durchschnitten”, Math. Ann., 255 (1977), 161–171 | DOI | MR

[33] Siu Y.-T., Trautmann G., Gap-sheaves and extension of coherent analytic subsheaves, Lecture Notes in Math., No 172, Springer-Verlag, 1971 | MR

[34] Steenbrink J., “Intersection form for quasi-homogeneous singularities”, Compos. Math., 34:2 (1977), 211–223 | MR | Zbl

[35] Steenbrink J., “Mixed Hodge structure on the vanishing cohomology”, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, 525–563 | MR

[36] Tyurina G. N., “O topologicheskikh svoistvakh izolirovannykh osobennostei kompleksnykh prostranstv korazmernosti odin”, Izv. AN SSSR. Ser. matem., 32:3 (1968), 605–620 | Zbl

[37] Wagreich P., “Algebras of automorphic forms with few generators”, Transactions Amer. Math. Soc., 262:2 (1980), 367–389 | DOI | MR | Zbl

[38] Wahl J., Derivations and deformations of quasi-homogeneous singularities, preprint Arcata-81, Berkley, California, 1982

[39] Wahl J., “Derivations of negative weight and nonsmoothability of certain singularities”, Math. Ann., 258 (1982), 383–398 | DOI | MR | Zbl

[40] Wall C. T. C., “A second note on symmetry of singularities”, Bull. London Math. Soc., 12:5 (1980), 347–354 | DOI | MR | Zbl

[41] Aleksandrov A. G., “O deformatsiyakh buketov kvaziodnorodnykh odnomernykh osobennostei”, Funkts. analiz i ego prilozheniya, 15:1 (1981), 67–68 | MR | Zbl

[42] Aleksandrov A. G., “Normalnye formy odnomernykh kvaziodnorodnykh polnykh peresechenii”, Matem. sb., 117:1 (1982), 3–31 | MR | Zbl

[43] Aleksandrov A. G., “O komplekse de Rama kzaziodnorodnogo polnogo peresecheniya”, Funkts. analiz i ego prilozheniya, 17:1 (1983), 63–64 | MR | Zbl

[44] Aleksandrov A. G., “Vektornye polya i deformatsii kvaziodnorodnogo polnogo peresecheniya”, XVII vsesoyuznaya algebraicheskaya konferentsiya (tez. soobschenii, ch. I), Minsk, 1983, 6

[45] Retakh V. S., “Operatsii Massi v superalgebrakh Li i deformatsii kompleksno-analiticheskikh algebr”, Funkts. analiz i ego prilozheniya, 11:4 (1977), 88–89 | MR | Zbl

[46] Laudal O. A., Pfister G., The local moduli problem, applications to the classification of isolated singularities, preprint Institute of Math. Univ., Oslo, 1983

[47] Wahl J. M., “Derivations, automorphisms and deformations of quasi-homogeneous singularities”, Singularities, part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983, 613–624 | MR