Singularities of Narasimhan-Simha type metrics on direct images of relative pluricanonical bundles  [ Singularités des métriques de type Narasimhan-Simha sur l’image directe du fibré pluricanonique relatif ]
Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 753-783.

Nous étudions les singularités des métriques hermitiennes de Narasimhan-Simha sur les images directes des fibrés pluricanoniques relatifs. La majoration est liée aux seuils log-canoniques.

We study singularities of the Narasimhan-Simha Hermitian metric on the direct image of a relative pluricanonical bundle. The upper bound relates to log-canonical thresholds.

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Classification : 14D06,  32J25
Mots clés : Singularités des métriques de Narasimhan-Simha, fibré pluricanonique relatif, métrique du noyau de Bergman, fibré vectoriel nef
     author = {Takayama, Shigeharu},
     title = {Singularities of Narasimhan-Simha type metrics on direct images of relative pluricanonical bundles},
     journal = {Annales de l'Institut Fourier},
     pages = {753--783},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {2},
     year = {2016},
     doi = {10.5802/aif.3025},
     language = {en},
     url = {}
Takayama, Shigeharu. Singularities of Narasimhan-Simha type metrics on direct images of relative pluricanonical bundles. Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 753-783. doi : 10.5802/aif.3025.

[1] Abramovich, D.; Karu, K. Weak semistable reduction in characteristic 0, Invent. Math., Tome 139 (2000) no. 2, pp. 241-273 | Article

[2] Berndtsson, Bo; Păun, Mihai Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J., Tome 145 (2008) no. 2, pp. 341-378 | Article

[3] Berndtsson, Bo; Păun, Mihai Bergman kernels and subadjunction (2010) (

[4] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., Tome 23 (2010) no. 2, pp. 405-468 | Article

[5] Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties, Graduate Studies in Mathematics, Tome 124, American Mathematical Society, Providence, RI, 2011, xxiv+841 pages | Article

[6] Demailly, Jean-Pierre Regularization of closed positive currents and intersection theory, J. Algebraic Geom., Tome 1 (1992) no. 3, pp. 361-409

[7] Demailly, Jean-Pierre Analytic methods in algebraic geometry, Surveys of Modern Mathematics, Tome 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012, viii+231 pages

[8] Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math., Tome 12 (2001) no. 6, pp. 689-741 | Article

[9] Fornæss, John Erik; Narasimhan, Raghavan The Levi problem on complex spaces with singularities, Math. Ann., Tome 248 (1980) no. 1, pp. 47-72 | Article

[10] Fujino, Osamu Direct images of relative pluricanonical bundles (to appear in Algebr. Geom.,

[11] Fujita, Takao On Kähler fiber spaces over curves, J. Math. Soc. Japan, Tome 30 (1978) no. 4, pp. 779-794 | Article

[12] Griffiths, Phillip A. Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Inst. Hautes Études Sci. Publ. Math. (1970) no. 38, pp. 125-180 | Article

[13] Hacon, Christopher D.; Xu, Chenyang Existence of log canonical closures, Invent. Math., Tome 192 (2013) no. 1, pp. 161-195 | Article

[14] Kawamata, Yujiro Characterization of abelian varieties, Compositio Math., Tome 43 (1981) no. 2, pp. 253-276

[15] Kawamata, Yujiro Kodaira dimension of algebraic fiber spaces over curves, Invent. Math., Tome 66 (1982) no. 1, pp. 57-71 | Article

[16] Kawamata, Yujiro Subadjunction of log canonical divisors for a subvariety of codimension 2, Birational algebraic geometry (Baltimore, MD, 1996) (Contemp. Math.) Tome 207, Amer. Math. Soc., Providence, RI, 1997, pp. 79-88 | Article

[17] Kawamata, Yujiro Subadjunction of log canonical divisors. II, Amer. J. Math., Tome 120 (1998) no. 5, pp. 893-899 | Article

[18] Kobayashi, Shoshichi Geometry of bounded domains, Trans. Amer. Math. Soc., Tome 92 (1959), pp. 267-290 | Article

[19] Kollár, János Subadditivity of the Kodaira dimension: fibers of general type, Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.) Tome 10, North-Holland, Amsterdam, 1987, pp. 361-398

[20] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Tome 134, Cambridge University Press, Cambridge, 1998, viii+254 pages (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | Article

[21] Lazarsfeld, Robert Positivity in algebraic geometry. I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Tome  48, 49, Springer-Verlag, Berlin, 2004 | Article

[22] Mourougane, Christophe; Takayama, Shigeharu Extension of twisted Hodge metrics for Kähler morphisms, J. Differential Geom., Tome 83 (2009) no. 1, pp. 131-161

[23] Narasimhan, M. S.; Simha, R. R. Manifolds with ample canonical class, Invent. Math., Tome 5 (1968), pp. 120-128 | Article

[24] Okonek, Christian; Schneider, Michael; Spindler, Heinz Vector bundles on complex projective spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2011, viii+239 pages (Corrected reprint of the 1988 edition, With an appendix by S. I. Gelfand)

[25] Păun, Mihai Siu’s invariance of plurigenera: a one-tower proof, J. Differential Geom., Tome 76 (2007) no. 3, pp. 485-493

[26] Păun, Mihai; Takayama, Shigeru Positivity of twisted relative pluricanonical bundles and their direct images (

[27] Popa, Mihnea; Schnell, Christian On direct images of pluricanonical bundles (

[28] Raufi, Hossein Singular hermitian metrics on holomorphic vector bundles (

[29] Sakai, F. Kodaira dimensions of complements of divisors, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 239-257

[30] Siu, Yum-Tong Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 223-277

[31] Takayama, Shigeharu On the invariance and the lower semi-continuity of plurigenera of algebraic varieties, J. Algebraic Geom., Tome 16 (2007) no. 1, pp. 1-18 | Article

[32] Tsuji, Hajime Curvature semipositivity of relative pluricanonical systems (

[33] Viehweg, Eckart Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 30, Springer-Verlag, Berlin, 1995, viii+320 pages | Article

[34] Yoshikawa, Ken-Ichi Singularities and analytic torsion (