00:00:00 / 00:00:00

Positivity of higher exterior powers of the tangent bundle

By Cécile Gachet

Appears in collection : Algebraic geometry and complex geometry / Géométrie algébrique et géométrie complexe

Positivity of the tangent bundle and its top exterior power (namely, the anticanonical bundle) is the subject of extensive literature, and several open problems. Notably, Campana and Peternell predict that if $-K_{X}$ is strictly nef, then $X$ is a Fano variety: this conjecture is proven up to dimension 3 by the work of Maeda and Serrano. In this talk, we investigate positivity of the intermediate exterior powers of the tangent bundle. We prove that if $X$ is a smooth projective n-fold and the third, fourth or (n-1)-th exterior power of $T_{X}$ is strictly nef, then $X$ is a Fano variety. Moreover, we classify smooth projective varieties of Picard number at least two with third or fourth exterior power of $T_{X}$ strictly nef. This work is actually slightly more general, as it boils down to classifying rationally connected varieties such that the degree of the anticanonical bundle on rational curves is quite large.

Information about the video

Citation data

  • DOI 10.24350/CIRM.V.19983403
  • Cite this video Gachet Cécile (12/1/22). Positivity of higher exterior powers of the tangent bundle. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19983403
  • URL https://dx.doi.org/10.24350/CIRM.V.19983403



Last related questions on MathOverflow

You have to connect your Carmin.tv account with mathoverflow to add question

Ask a question on MathOverflow


  • Bookmark videos
  • Add videos to see later &
    keep your browsing history
  • Comment with the scientific
  • Get notification updates
    for your favorite subjects
Give feedback