The direct image of the relative dualizing sheaf needs not be semiample. (L’image directe du faisceau dualisant relatif n’est pas nécessairement semi-ample.) (English. French summary) Zbl 1310.14017
The authors provide details for the proof of Fujita’s second theorem [T. Fujita, Proc. Japan Acad., Ser. A 54, 183–184 (1978; Zbl 0412.32029)] claiming that if a fibration \(f: X\to B\) of a compact Kähler manifold \(X\) over a projective curve \(B\) is given, then the direct image sheaf \(V:=f_*\omega_{X/B}\) splits as a direct sum \(V=A\oplus Q\), where \(A\) is an ample vector bundle and \(Q\) is a unitary flat bundle. The main result is Theorem 1.2, claiming that \(V\) needs not be semiample; this theorem answers the question posed by Fujita in 1982.
Reviewer: Vladimir L. Popov (Moskva)
MSC:
| 14D06 | Fibrations, degenerations in algebraic geometry |
| 32L05 | Holomorphic bundles and generalizations |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 32Q15 | Kähler manifolds |
Citations:
Zbl 0412.32029References:
| [1] | Catanese, F.; Dettweiler, M., Answer to a question by Fujita on variation of Hodge structures (2013), preprint, 26 pages |
| [2] | Deligne, P.; Mostow, G. D., Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. IHÉS, 63, 5-89 (1986) · Zbl 0615.22008 |
| [3] | Fujita, Takao, On Kähler fiber spaces over curves, J. Math. Soc. Jpn., 30, 4, 779-794 (1978) · Zbl 0393.14006 |
| [4] | Fujita, Takao, The sheaf of relative canonical forms of a Kähler fiber space over a curve, Proc. Jpn. Acad., Ser. A, Math. Sci., 54, 7, 183-184 (1978) · Zbl 0412.32029 |
| [5] | Griffiths, P., Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Publ. Math. IHÉS, 38, 125-180 (1970) · Zbl 0212.53503 |
| [6] | Griffiths, P., Topics in Transcendental Algebraic Geometry, Annals of Mathematics Studies, vol. 106 (1984), Princeton University Press · Zbl 0528.00004 |
| [7] | Griffiths, P.; Harris, J., Principles of Algebraic Geometry, Pure and Applied Mathematics (1978), Wiley-Interscience: Wiley-Interscience New York · Zbl 0408.14001 |
| [8] | Griffiths, P.; Schmid, W., Recent developments in Hodge theory: A discussion of techniques and results, (Discrete Subgroups of Lie Groups Appl. Moduli, Pap. Bombay Colloq.. Discrete Subgroups of Lie Groups Appl. Moduli, Pap. Bombay Colloq., 1973 (1975)), 31-127 · Zbl 0355.14003 |
| [9] | Hartshorne, R., Ample vector bundles on curves, Nagoya Math. J., 43, 73-89 (1971) · Zbl 0218.14018 |
| [10] | Classification of algebraic and analytic manifolds, (Ueno, Kenji, Proc. Symp. Katata/Jap.. Proc. Symp. Katata/Jap., 1982. Proc. Symp. Katata/Jap.. Proc. Symp. Katata/Jap., 1982, Progress in Mathematics, vol. 39 (1983), Birkhäuser: Birkhäuser Boston, Mass.), 591-630, Open problems: Classification of algebraic and analytic manifolds · Zbl 0527.14002 |
| [11] | Kawamata, Y., Kodaira dimension of algebraic fiber spaces over curves, Invent. Math., 66, 1, 57-71 (1982) · Zbl 0461.14004 |
| [12] | Kempf, G.; Knudsen, F. F.; Mumford, D.; Saint Donat, B., Toroidal Embeddings, I, Lecture Notes in Mathematics, vol. 739 (1973), Springer, viii+209 p · Zbl 0271.14017 |
| [13] | Kohno, M., Global Analysis in Linear Differential Equations (1999), Kluwer Academic Publishers · Zbl 0933.34002 |
| [14] | Kollár, J., Higher direct images of dualizing sheaves. I, II, Ann. Math. (2). Ann. Math. (2), Ann. Math. (2), 124, 171-202 (1986) · Zbl 0605.14014 |
| [15] | Lazarsfeld, R., Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, vol. 49 (2004), Springer-Verlag: Springer-Verlag Berlin, xviii+385 p · Zbl 1093.14500 |
| [16] | Peters, C. A.M., A criterion for flatness of Hodge bundles over curves and geometric applications, Math. Ann., 268, 1, 1-19 (1984) · Zbl 0548.14004 |
| [17] | Schmid, W., Variation of Hodge structure: The singularities of the period mapping, Invent. Math., 22, 211-319 (1973) · Zbl 0278.14003 |
| [18] | Schwarz, H. A., Über diejenigen Fälle in welchen die Gaussische hypergeometrische Reihe eine algebraische Funktion ihres vierten Elements darstellt, J. Reine Angew. Math., 75, 292-335 (1873) · JFM 05.0146.03 |
| [19] | Zucker, S., Hodge theory with degenerating coefficients: \(L^2\)-cohomology in the Poincaré metric, Ann. Math. (2), 109, 415-476 (1979) · Zbl 0446.14002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
