Stability of tangent bundles of complete intersections and effective restriction

Abstract

For $n\ge 3$, let $M$ be an $(n+r)$-dimensional irreducible Hermitian symmetric space of compact type and let ${\mathcal{O}}_M(1)$ be the ample generator of $Pic(M)$. Let $Y=H_1\cap \cdots \cap H_r$ be a smooth complete intersection of dimension $n$ where $H_i\in |{\mathcal{O}}_M(d_i)|$ 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 $Pic(Y)\to Pic(X)$ 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 ${\mathbb{P}}^{n+1}$ and $X$ is 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.

The paper is significantly rewritten. More results on complete intersections in Hermitian symmetric spaces are included. Any comments are welcome

References in corpus (1)

• On stability of tangent bundles of Fano manifolds with b_2=1