Linearization of transition functions of a semi-positive line bundle along a certain submanifold

Koike, Takayuki

Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 2237-2271.

Soit $X$ une variété complexe et $L$ un fibré en droites holomorphe sur $X$. Supposons que $L$ est semi-positive, à savoir $L$ admet une métrique hermitienne lisse avec une courbure de Chern semi-positif. Soit $Y$ une sous-variété kählérienne compacte de $X$ telle que la restriction de $L$ à $Y$ est topologiquement triviale. Nous examinons l’obstruction pour que $L$ soit plat unitaire sur un voisinage de $Y$. Comme application, par exemple, nous prouvons l’existence d’un fibré en droites nef, grand et non semi-positif sur une surface projective non singulière.

Let $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$. Assume that $L$ is semi-positive, namely $L$ admits a smooth Hermitian metric with semi-positive Chern curvature. Let $Y$ be a compact Kähler submanifold of $X$ such that the restriction of $L$ to $Y$ is topologically trivial. We investigate the obstruction for $L$ to be unitary flat on a neighborhood of $Y$ in $X$. As an application, for example, we show the existence of nef, big, and non semi-positive line bundle on a non-singular projective surface.

Classification : 32J25, 14C20
Keywords: Hermitian metrics, neighborhoods of subvarieties, Ueda theory
Mot clés : métrique hermitienne, voisinage de sous-variété, théorie d’Ueda

[1] Arnol’d, Vladimir I. Bifurcations of invariant manifolds of differential equations and normal forms in neighborhoods of elliptic curves, Funct. Anal. Appl., Volume 10 (1977) no. 4, pp. 249-259 translation from Funkts. Anal. Prilozh. 10, no. 4, p. 1-12 (1976) | Zbl

[2] Bedford, Eric; Kalka, Morris Foliations and complex Monge–Ampère equations, Commun. Pure Appl. Math., Volume 30 (1977) no. 5, pp. 543-571 | DOI | Zbl

[3] Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | DOI | Zbl

[4] Brunella, Marco On Kähler surfaces with semipositive Ricci curvature, Riv. Mat. Univ. Parma, Volume 1 (2010) no. 2, pp. 441-450 | Zbl

[5] Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael Compact complex manifolds with numerically effective tangent bundles, J. Algebr. Geom., Volume 3 (1994) no. 2, pp. 295-345 | MR | Zbl

[6] Filip, Simion; Tosatti, Valentino Smooth and Rough Positive Currents, Ann. Inst. Fourier, Volume 68 (2018) no. 7, pp. 2981-2999 | DOI | Numdam | MR | Zbl

[7] Kim, Dano Extension of pluriadjoint sections from a log-canonical center, Ph. D. Thesis, Princeton University, Princeton, USA (2007) | MR

[8] Koike, Takayuki On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated, Ann. Inst. Fourier, Volume 65 (2015) no. 5, pp. 1953-1967 | DOI | Numdam | MR | Zbl

[9] Koike, Takayuki On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles, Kyoto J. Math., Volume 55 (2015) no. 3, pp. 607-616 | MR | Zbl

[10] Koike, Takayuki Higher codimensional Ueda theory for a compact submanifold with unitary flat normal bundle, Nagoya Math. J., Volume 238 (2018), pp. 104-136 | DOI | MR | Zbl

[11] Koike, Takayuki Hermitian metrics on the anti-canonical bundle of the blow-up of the projective plane at nine points (2019) (https://arxiv.org/abs/1909.06827)

[12] Koike, Takayuki; Ogawa, Noboru On the neighborhood of a torus leaf and dynamics of holomorphic foliations (2020) (https://arxiv.org/abs/1808.10219)

[13] Neeman, Amnon Ueda theory: theorems and problems, Memoirs of the American Mathematical Society, 415, American Mathematical Society, 1989 | Zbl

[14] Ohsawa, Takeo Vanishing theorems on complete Kähler manifolds, Publ. Res. Inst. Math. Sci., Volume 20 (1984) no. 1, pp. 21-38 | DOI | Zbl

[15] Sommer, Friedrich Komplex-analytische Blätterung reeller Hyperflächen im $C^n$, Math. Ann., Volume 137 (1959) no. 5, pp. 392-411 | DOI | MR | Zbl

[16] Ueda, Tetsuo On the neighborhood of a compact complex curve with topologically trivial normal bundle, J. Math. Kyoto Univ., Volume 22 (1983), pp. 583-607 | MR | Zbl