Hodge metrics and the curvature of higher direct images
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 6, p. 905-924

Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct image bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.

Nous utilisons la théorie de représentation par formes harmoniques des classes de cohomologie relative développée par Takegoshi et la structure des calculs de courbure de fibrés images directes développée par Berndtsson, pour étudier les images directes supérieures par un morphisme lisse du fibré canonique relatif tensorisé par un fibré vectoriel holomorphe hermitien semi-positif. Nous montrons qu'elles sont localement libres et que, munies de métriques convenables de type Hodge, elles sont à courbure semi-positive.

DOI : https://doi.org/10.24033/asens.2084
Classification:  32L10,  14F10,  32J25,  14D07
Keywords: higher direct images, Hodge metrics, harmonic theory for relative cohomology, Nakano positivity
@article{ASENS_2008_4_41_6_905_0,
     author = {Mourougane, Christophe and Takayama, Shigeharu},
     title = {Hodge metrics and the curvature of higher direct images},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {6},
     year = {2008},
     pages = {905-924},
     doi = {10.24033/asens.2084},
     zbl = {1167.14027},
     mrnumber = {2504108},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2008_4_41_6_905_0}
}
Mourougane, Christophe; Takayama, Shigeharu. Hodge metrics and the curvature of higher direct images. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 6, pp. 905-924. doi : 10.24033/asens.2084. http://www.numdam.org/item/ASENS_2008_4_41_6_905_0/

[1] C. Bănică & O. Stănăasila, Algebraic methods in the global theory of complex spaces, Editura Academiei, 1976. | MR 463470 | Zbl 0334.32001

[2] B. Berndtsson, Curvature of vector bundles associated to holomorphic fibrations, preprint arXiv:math.CV/0511225v2, to appear in Ann. of Math. | MR 2480611 | Zbl 1195.32012

[3] B. Berndtsson & M. Păun, Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math J. 145 (2008), 341-378. | MR 2449950 | Zbl 1181.32025

[4] I. Enoki, Kawamata-Viehweg vanishing theorem for compact Kähler manifolds, in Einstein metrics and Yang-Mills connections (Sanda, 1990), Lecture Notes in Pure and Appl. Math., 145, Dekker, 1993, 59-68. | MR 1215279 | Zbl 0797.53052

[5] H. Esnault & E. Viehweg, Lectures on vanishing theorems, DMV Seminar, 20, Birkhäuser, 1992. | MR 1193913 | Zbl 0779.14003

[6] O. Fujino, Higher direct images of log canonical divisors, J. Differential Geom. 66 (2004), 453-479. | MR 2106473 | Zbl 1072.14019

[7] T. Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), 779-794. | MR 513085 | Zbl 0393.14006

[8] H. Grauert & R. Remmert, Coherent analytic sheaves, Grund. Math. Wiss., 265, Springer, 1984. | MR 755331 | Zbl 0537.32001

[9] P. A. Griffiths, Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Publ. Math. I.H.É.S. 38 (1970), 125-180. | Numdam | MR 282990 | Zbl 0212.53503

[10] R. Hartshorne, Algebraic geometry, Graduate Texts in Math., 52, Springer, 1977. | MR 463157 | Zbl 0367.14001

[11] D. Huybrechts, Complex geometry, Universitext, Springer, 2005. | MR 2093043 | Zbl 1055.14001

[12] Y. Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), 253-276. | Numdam | MR 622451 | Zbl 0471.14022

[13] Y. Kawamata, Subadjunction of log canonical divisors. II, Amer. J. Math. 120 (1998), 893-899. | MR 1646046 | Zbl 0919.14003

[14] K. Kodaira, Complex manifolds and deformation of complex structures, Grund. Math. Wiss., 283, Springer, 1986. | MR 815922 | Zbl 0581.32012

[15] J. Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. 123 (1986), 11-42. | MR 825838 | Zbl 0598.14015

[16] J. Kollár, Higher direct images of dualizing sheaves. II, Ann. of Math. 124 (1986), 171-202. | MR 847955 | Zbl 0605.14014

[17] J. Kollár, Kodaira's canonical bundle formula and adjunction, in Flips for 3-folds and 4-folds (A. Corti, éd.), 2007. | MR 2359346 | Zbl 1286.14027

[18] N. Levenberg & H. Yamaguchi, The metric induced by the Robin function, Mem. Amer. Math. Soc. 92 (1991), 156. | MR 1061928 | Zbl 0742.31003

[19] F. Maitani & H. Yamaguchi, Variation of Bergman metrics on Riemann surfaces, Math. Ann. 330 (2004), 477-489. | MR 2099190 | Zbl 1077.32006

[20] L. Manivel, Un théorème de prolongement L 2 de sections holomorphes d’un fibré hermitien, Math. Z. 212 (1993), 107-122. | MR 1200166 | Zbl 0789.32015

[21] S. Mori, Classification of higher-dimensional varieties, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Amer. Math. Soc., 1987, 269-331. | MR 927961 | Zbl 0656.14022

[22] A. Moriwaki, Torsion freeness of higher direct images of canonical bundles, Math. Ann. 276 (1987), 385-398. | MR 875335 | Zbl 0589.14016

[23] C. Mourougane, Images directes de fibrés en droites adjoints, Publ. Res. Inst. Math. Sci. 33 (1997), 893-916. | MR 1614576 | Zbl 0926.14004

[24] C. Mourougane & S. Takayama, Hodge metrics and positivity of direct images, J. reine angew. Math. 606 (2007), 167-178. | MR 2337646 | Zbl 1128.14030

[25] T. Ohsawa, On the extension of L 2 holomorphic functions. II, Publ. Res. Inst. Math. Sci. 24 (1988), 265-275. | MR 944862 | Zbl 0653.32012

[26] T. Ohsawa & K. Takegoshi, On the extension of L 2 holomorphic functions, Math. Z. 195 (1987), 197-204. | MR 892051 | Zbl 0625.32011

[27] K. Takegoshi, Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kähler morphisms, Math. Ann. 303 (1995), 389-416. | MR 1354997 | Zbl 0843.32018

[28] H. Tsuji, Variation of Bergman kernels of adjoint line bundles, preprint arXiv:math.CV/0511342.

[29] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, in Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1 (1983), 329-353. | MR 715656 | Zbl 0513.14019

[30] E. Viehweg, Quasi-projective moduli for polarized manifolds, Ergeb. Math. und ihrer Grenzgebiete, 30, Springer, 1995. | MR 1368632 | Zbl 0844.14004

[31] C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés, 10, Soc. Math. de France, 2002, version anglaise : Hodge Theory and Complex algebraic geometry I and II, Cambridge Studies in advanced Mathematics 76 et 77. | MR 2744215 | Zbl 1032.14001

[32] H. Yamaguchi, Variations of pseudoconvex domains over n , Michigan Math. J. 36 (1989), 415-457. | MR 1027077 | Zbl 0692.31004