Geometry and Analysis Seminar


On Fano manifolds with

numerically effective tangent bundle


Professor Ngaiming Mok

The University of Hong Kong


A rational homogeneous space is a projective manifold on which a semisimple Lie group acts transitively. Among them the class of Hermitian symmetric spaces of compact type is most familiar. They include among other things projective spaces,  hyperquadrics and Grassmann manifolds. After fundamental works of Mori and  Siu-Yau, we resolved in 1987 the Generalized Frankel Conjecture in Kähler  Geometry, proving in particular that a Fano manifold X admitting a Kähler metric  of semipositive holomorphic bisectional curvature is biholomorphic to a Hermitian symmetric manifold of compact type. In view of Mori's famous solution to the Hartshorne Conjecture in 1979, one expects that there is an analogue to the Generalized Frankel Conjecture in Algebraic Geometry. In fact, Campana-Peternell formulated such a conjecture in 1991, replacing the semipositivity assumption in terms of Kähler metrics by the algebro-geometric hypothesis that the tangent bundle of X is numerically effective, in which case X is conjectured to be rational homogenenous. The fundamental case of the Conjecture is for the class of Fano manifolds of Picard number 1. We contribute to a very special case of the Conjecture, viz. when b2(X) = b4(X) = 1 and the varieties of minimal rational curves are 1-dimensional. For the proof we make use of the stability of certain rank-2 vector bundles constructed on the moduli space of minimal rational curves.

Lecture I:

September 20, 2000 (Wed)

4:00 - 5:00pm

Lecture II:

September 27, 2000 (Wed)

4:00 - 5:00pm

Lecture III:

October 18, 2000 (Wed)

4:00 - 5:00pm

Lecture IV:

October 25, 2000 (Wed)

4:00 - 5:00pm

Lecture V:

November 22, 2000 (Wed)

4:00 - 5:00pm


Lectures will be held in Room 517, Meng Wah Complex

All are welcome