Stability of tangent bundles of complete intersections and effective restriction

TL;DR

This paper proves a vanishing theorem for twisted holomorphic forms on complete intersections in Hermitian symmetric spaces, establishing the stability of tangent bundles and their restrictions under certain conditions.

Contribution

It introduces new vanishing results and stability criteria for tangent bundles of complete intersections and their hypersurface restrictions in Hermitian symmetric spaces.

Findings

T_Y's tangent bundle is stable for smooth complete intersections.

Effective bounds are provided for the degree d ensuring stability of T_Y|_X.

Stability of T_Y|_X holds for general hypersurfaces in projective space, with known exceptions.

Abstract

For $n\geq 3$, 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 $Y=H_1\cap\dots\cap H_r$ be a smooth complete intersection of dimension $n$ where $H_i\in\vert \mathcal{O}_M(d_i)\vert$ with $d_i\geq 2$. 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 $Pic(Y)\rightarrow Pic(X)$ is surjective, we establish some effective results for $d$ to guarantee the stability of the restriction $T_Y\vert_X$. In particular, if $Y$ is a general hypersurface in $\mathbb{P}^{n+1}$ and $X$ is general smooth divisor in $Y$, we show that $T_Y\vert_X$ is stable except for some well-known examples. We also address the cases where the Picard group increases by restriction.