# ANNALES DE L'INSTITUT FOURIER

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 $n\ge 3$ and $r\ge 1$, let $M$ be an $\left(n+r\right)$-dimensional irreducible Hermitian symmetric space of compact type and let ${𝒪}_{M}\left(1\right)$ be the ample generator of $\mathrm{pic}\left(M\right)$. Let $Y={H}_{1}\cap \cdots \cap {H}_{r}$ be a smooth complete intersection of dimension $n$, where ${H}_{i}\in |{𝒪}_{M}\left({d}_{i}\right)|$ with ${d}_{i}\ge 2$. 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 $\mathrm{pic}\left(Y\right)\to \mathrm{pic}\left(X\right)$ 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 ${ℙ}^{n+1}$ 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 $\left(n+r\right)$ avec $n\ge 3$ et $r\ge 1$, ${𝒪}_{M}\left(1\right)$ le générateur ample de $\mathrm{pic}\left(M\right)$. Soit $Y={H}_{1}\cap \cdots \cap {H}_{r}$ une intersection complète lisse de dimension $n$${H}_{i}\in |{𝒪}_{M}\left({d}_{i}\right)|$ avec ${d}_{i}\ge 2$. Nous montrons un théorème d’annulation pour le faisceau tordu des germes de $p$-formes holomorphes ${\Omega }_{Y}^{p}\left(\ell \right)$. 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 $\mathrm{pic}\left(Y\right)\to \mathrm{pic}\left(X\right)$ 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 ${ℙ}^{n+1}$ 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
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. https://aif.centre-mersenne.org/articles/10.5802/aif.3435/

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