Hodge metrics and the curvature of higher direct images

Mourougane, Christophe; Takayama, Shigeharu

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 6, pp. 905-924.

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.

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.

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

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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

[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. | EuDML | MR | Zbl

[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 | Zbl

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

[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 | Zbl

[32] H. Yamaguchi, Variations of pseudoconvex domains over ${ℂ}^n$, Michigan Math. J. 36 (1989), 415-457. | MR | Zbl