Instability of Truncated Symmetric Powers of sheaves

2010/10/20 by Lingguang Li, Fei Yu, Li, Lingguang +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Nonlinear Waves and Solitons #math.AG #msc:14H60 #msc:14J60

paper · pdf · doi:10.48550/arxiv.1010.4228

12 pages

arxiv created 2012/01/01 · arxiv updated 2012/01/04

Abstract

Let X be a smooth projective variety of dimension n over an algebraically closed field k of characteristic p>0. Let F_X:X→ X be the absolute Frobenius morphism, and \E a torsion free sheaf on X. We give a upper bound of instability of truncated symmetric powers T^l(\E)(0≤ l≤\rk(\E)(p-1)) in terms of L_max(\Omg¹_X), I(\Omg¹_X) and I(\E) (Theorem \refInstabTl). As an application, We obtain a upper bound of Frobenius direct image F_X_*(\E) and some sufficient conditions of slope semi-stability of F_X_*(\E). In addition, we study the slope (semi)-stability of sheaves of locally exact (closed) forms Bⁱ_X (Zⁱ_X).

Related

• Stability of sheaves of locally closed and exact forms
• Deciding Stability of Sheaves on Curves
• Stability of Frobenius direct images over surfaces
• Restriction theorems for semistable sheaves
• Stability of direct images under Frobenius morphism