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    • Keywords


      Semistable sheaf; tangent bundle; toric variety; Hirzebruch surface; Fano manifold.

    • 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).

    • Author Affiliations



      1. School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400 005, India
      2. Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India
      3. Middle East Technical University, Northern Cyprus Campus, Guzelyurt, Mersin 10, Turkey
      4. Present Address: Department of Mathematics and Computer Science, Jagiellonian University, ul. prof. Stanisława, Łojasiewicza 6, 30-348 Kraków, Poland
      5. Mathematics Department, Indian Institute of Science Education and Research, Pune 411 008, India
    • Dates

  • Proceedings – Mathematical Sciences | News

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