On stability of tangent bundle of toric varieties


Biswas I., Dey A., Genc O., Poddar M.

PROCEEDINGS OF THE INDIAN ACADEMY OF SCIENCES-MATHEMATICAL SCIENCES, vol.131, no.2, 2021 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 131 Issue: 2
  • Publication Date: 2021
  • Doi Number: 10.1007/s12044-021-00623-w
  • Title of Journal : PROCEEDINGS OF THE INDIAN ACADEMY OF SCIENCES-MATHEMATICAL SCIENCES
  • Keywords: Semistable sheaf, tangent bundle, toric variety, Hirzebruch surface, Fano manifold, SHEAVES

Abstract

Let X be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle TX. 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).