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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Varieties of minimal rational tangents and associated geometric substructures.

## Varieties of minimal rational tangents and associated geometric substructures.Add to your list(s) Download to your calendar using vCal - Ngaiming Mok (University of Hong Kong)
- Wednesday 11 September 2024, 11:30-12:30
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact nobody. TWTW01 - Twistors in Geometry & Physics 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. This talk is part of the Isaac Newton Institute Seminar Series series. ## This talk is included in these lists:- All CMS events
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