Stability of tangent bundles of complete intersections and effective restriction

2017/11/09 by Jie Liu, Liu, Jie · 1 citation
Mathematics · #14J70 #14M10 #32M15 #32M25 #32Q26 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1711.03413

openalex publication_date 2017/11/09 · openalex created_date 2017/11/17 · openalex updated_date 2026/07/28

Abstract

For n≥ 3, 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=H₁∩…∩ H_r be a smooth complete intersection of dimension n where H_i∈\vert O_M(d_i)\vert 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\vert_X. In particular, if Y is a general hypersurface in ℙⁿ⁺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.

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• Deformations of the tangent bundle of a projective hypersurface

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