Abstract

Let $X$ be a uniruled projective manifold.  Starting in the late 1990s, I have developed with Jun-Muk Hwang the basics of a geometric theory of varieties of minimal rational tangents (VMRTs) on $X$ which generalizes $S$-structures on irreducible Hermitian symmetric manifolds $S = G/P$.  One possible link of VMR Ts to twistor theory is the LeBrun-Salamon conjecture, according to which a compact quaternion-K\”ahler manifold $Q$ of positive scalar curvature is necessarily Riemannian symmetric.  One approach to confirming the conjecture is to consider the twistor space $Z$ associated to $Q$, which is known to be a Fano contact manifold. By the solution of the {\it Recognition Problem\/} in VMRT theory it is known that a Fano contact manifold of Picard number 1 is necessarily rational homogeneous, provided that the VMRT at a general point agrees with the VMRT of a contact homogeneous manifold of Picard number 1.  It is known that the VMRT $\mathscr C_x(Z)$ at a general point $x \in Z$ is an immersed Legendrian submanifold of $\mathbb PD_x$, where $D$ denotes the holomorphic contact distribution.  It remains however a difficult problem to identify $\mathbb PD_x$ at a general point. \vskip 0.3cm This lecture will focus on VMRT theory itself, and especially on the problem of characterizing a rational homogeneous manifold $X$ of Picard number 1 by its VMRT $\mathscr C_x(X) \subset \mathbb PT_x(X)$ at a general point, and also the problem of characterizing certain projective subvarieties of $X$ such as Schubert cycles.  In general, we study the problem of characterizing uniruled projective subvarieties of $X$ by means of sub-VMRT structures, and give applications of such a study to rigidity problems in algebraic geometry, K\”ahler geometry and several complex variables.