Jeong-Seop Kim, Positivity of Tangent Bundles of Fano Threefolds

As well as the Hartshorne-Frankel conjecture on the ampleness of tangent bundle, it has been asked to characterize a smooth projective variety X whose tangent bundle T_X attains certain positivity, e.g., nefness, k-ampleness, or bigness. But for the ampleness, the complete answers are not known even within the class of smooth Fano varieties, only partial answers are known in the case of lower dimension or lower Picard number, some of which rely on classification theorems. On the bigness of T_X, the characterization has been done recently in the case of dimension 2 (Höring-Liu-Shao) and dimension 3 with Picard number 1 (Höring-Liu) using a special divisor on P(T_X), called the total dual VMRT. In this talk, I will briefly review the classification of Fano threefolds and the theory of total dual VMRT. Then I will introduce some criteria to determine the bigness of T_X, and announce a result on the bigness of T_X in the case of dimension 3 with higher Picard number. This is joint work with Hosung Kim and Yongnam Lee.