Stability of the tangent bundles of complete intersections and effective restriction
Annales de l'Institut Fourier, Volume 71 (2021) no. 4, pp. 1601-1634.

For n3 and r1, let M be an (n+r)-dimensional irreducible Hermitian symmetric space of compact type and let 𝒪 M (1) be the ample generator of pic(M). Let Y=H 1 H r be a smooth complete intersection of dimension n, where H i |𝒪 M (d i )| with d i 2. We prove a vanishing theorem for twisted holomorphic forms on Y. As an application, we show that the tangent bundle T Y of Y is stable. Moreover, if X is a smooth hypersurface of degree d in Y such that the restriction pic(Y)pic(X) is surjective, we establish some effective results for d to guarantee the stability of the restriction T Y | X . In particular, if Y is a general hypersurface in n+1 and X is a general smooth divisor in Y, we show that T Y | X is stable except for some well-known examples. We also address the cases where the Picard group increases by restriction.

Soient M un espace hermitien symétrique irréductible de type compact et de dimension (n+r) avec n3 et r1, 𝒪 M (1) le générateur ample de pic(M). Soit Y=H 1 H r une intersection complète lisse de dimension nH i |𝒪 M (d i )| avec d i 2. Nous montrons un théorème d’annulation pour le faisceau tordu des germes de p-formes holomorphes Ω Y p (). Comme application, nous montrons que le fibré tangent T Y de Y est stable. De plus, si X est une hypersurface lisse de degré d dans Y telle que la restriction pic(Y)pic(X) soit surjective, nous obtenons des estimations effectives liées à la stabilité de la restriction T Y | X . En particulier, si Y est une hypersurface générale dans n+1 et X est un diviseur général, nous montrons que T Y | X est stable sauf certains exemples bien connus. Nous considérons aussi le cas où le nombre de Picard augmente par restriction.

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DOI: 10.5802/aif.3435
Classification: 14M10, 14J70, 32M15, 32M25, 32Q26
Keywords: stability, tangent bundle, Lefschetz property, complete intersection, Hermitian symmetric space
Mot clés : stabilité, fibré tangent, propriété de Lefschetz, intersection complète, espace hermitien symétrique
Liu, Jie 1

1 Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing, 100190 (China)
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Liu, Jie. Stability of the tangent bundles of complete intersections and effective restriction. Annales de l'Institut Fourier, Volume 71 (2021) no. 4, pp. 1601-1634. doi : 10.5802/aif.3435. http://www.numdam.org/articles/10.5802/aif.3435/

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