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Beijing–Moscow Mathematics Colloquium
16 октября 2020 г. 13:30–14:30, г. Москва, online
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Characterizing homogeneous rational projective varieties with
Picard number 1 by their varieties of minimal rational tangents
D. A. Timashev Moscow State University
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Количество просмотров: |
Эта страница: | 150 |
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Аннотация:
It is well known that rational algebraic curves play a key role
in the geometry of complex projective varieties, especially of Fano
manifolds. In particular, on Fano manifolds of Picard number (= the 2nd
Betti number) one, which are sometimes called "unipolar", one may consider
rational curves of minimal degree passing through general points. Tangent
directions of minimal rational curves through a general point $x$ in a
unipolar Fano manifold $X$ form a projective subvariety
$\mathcal{C}_{x,X}$ in the projectivized tangent space $\mathbb{P}(T_xX)$,
called the variety of minimal rational tangents (VMRT).
In 90-s J.-M. Hwang and N. Mok developed a philosophy declaring that the
geometry of a unipolar Fano manifold is governed by the geometry of its
VMRT at a general point, as an embedded projective variety. In support of
this thesis, they proposed a program of characterizing unipolar flag
manifolds in the class of all unipolar Fano manifolds by their VMRT. In
the following decades a number of partial results were obtained by Mok,
Hwang, and their collaborators.
Recently the program was successfully completed (J.-M. Hwang, Q. Li, and
the speaker). The main result states that a unipolar Fano manifold $X$
whose VMRT at a general point is isomorphic to the one of a unipolar flag
manifold $Y$ is itself isomorphic to $Y$. Interestingly, the proof of the
main result involves a bunch of ideas and techniques from "pure" algebraic
geometry, differential geometry, structure and representation theory of
simple Lie groups and algebras, and theory of spherical varieties (which
extends the theory of toric varieties).
Язык доклада: английский
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