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SUMMARY:Varieties of minimal rational tangents and associated  geometric 
 substructures. - Ngaiming Mok (University of Hong Kong)
DTSTART:20240911T090000Z
DTEND:20240911T100000Z
UID:TALK219592@talks.cam.ac.uk
DESCRIPTION:Let $X$ be a uniruled projective manifold. &nbsp\;Starting in 
 the late 1990s\, I have developed with Jun-Muk Hwang the basics of a geome
 tric theory of varieties of minimal rational tangents (VMRTs) on $X$ which
  generalizes $S$-structures on irreducible Hermitian symmetric manifolds $
 S = G/P$. &nbsp\;One possible link of VMRTs to twistor theory is the LeBru
 n-Salamon conjecture\, according to which a compact quaternion-K\\"ahler m
 anifold $Q$ of positive scalar curvature is necessarily Riemannian symmetr
 ic. &nbsp\;One approach to confirming the conjecture is to consider the tw
 istor space $Z$ associated to $Q$\, which is known to be a Fano contact ma
 nifold. By the solution of the {\\it Recognition Problem\\/} in VMRT theor
 y it is known that a Fano contact manifold of Picard number 1 is necessari
 ly rational homogeneous\, provided that the VMRT at a general point agrees
  with the VMRT of a contact homogeneous manifold of Picard number 1. &nbsp
 \;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 $D$ de
 notes the holomorphic contact distribution. &nbsp\;It remains however a di
 fficult problem to identify $\\mathbb PD_x$ at a general point.\n\\vskip 0
 .3cm\nThis lecture will focus on VMRT theory itself\, and especially on th
 e 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 gen
 eral point\, and also the problem of characterizing certain projective sub
 varieties of $X$ such as Schubert cycles. &nbsp\;In general\, we study the
  problem of characterizing uniruled projective subvarieties of $X$ by mean
 s of sub-VMRT structures\, and give applications of such a study to rigidi
 ty problems in algebraic geometry\, K\\"ahler geometry and several complex
  variables.\n&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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