On stability of tangent bundle of toric varieties

Let X be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle T X . In particular, a complete answer is given when X is a Fano toric variety of dimension four with Picard number at most two, complementing the earlier work of Nakagawa ( Tohoku. Math. J. 45 (1993) 297–310; 46 (1994) 125–133). We also give an infinite set of examples of Fano toric varieties for which TX is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko ( Izv. Akad. Nauk. SSSR Ser. Mat. 53 (1989) 1001–1039, 1135) and developed further by Perling ( Math. Nachr . 263/264 (2004) 181–197) and Kool (Moduli spaces of sheaves on toric varieties, Ph.D. thesis (2010) (University of Oxford); Adv. Math . 227 (2011) 1700–1755).