Stability of the tangent bundles of complete intersections and effective restriction

Liu, Jie

Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1601-1634.

Soient $M$ un espace hermitien symétrique irréductible de type compact et de dimension $(n+r)$ 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 $n$ où $H_i\in \left|{𝒪}_M\left(d_i\right)\right|$ 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 ${\mathbb{P}}^{n+1}$ 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.

For $n\ge 3$ and $r\ge 1$, let $M$ be an $(n+r)$-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 $H_i\in \left|{𝒪}_M\left(d_i\right)\right|$ 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 ${\mathbb{P}}^{n+1}$ 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.

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

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