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Session 2. Algebraic Geometry
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Fano manifolds whose elementary contractions are smooth \({\bf P}^1\)-fibrations |
Jarosław A. Wiśniewski, Institute of Mathematics, University of Warsaw, Poland
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The talk is based on the joint work with Gianluca Occhetta
(Trento), Luis Sola Conde (Madrid) and Kiwamu Watanabe
Saitama
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This presentation concerns a geometric characterization of complete
flag varieties for semisimple algebraic groups. Namely, if \(X\) is
a Fano manifold whose all elementary contractions are \({\bf
P}^1\)-fibrations then \(X\) is isomorphic to the complete flag
manifold \(G/B\) where \(G\) is a semi-simple Lie algebraic group
and \(B\) is a Borel subgroup of \(G\).
Our proof of this statement is based on the following ideas: Every
smooth \({\bf P}^1\)-fibration of \(X\) provides an involution of
the vector space \(N^1(X)\) of classes of \({\bf R}\)-divisors in
\(X\). We show that these involutions generate a finite reflection
group, which is the Weyl group \(W\) of a semisimple Lie group
\(G\). Next we use \({\bf P}^1\)-fibrations of \(X\) to define a
set of auxiliary manifolds called Bott-Samelson varieties of \(X\),
which are analogues of the Bott-Samelson varieties that appear
classically in the study of Schubert cycles of flag
varieties. Subsequently we show that the recursive construction of
appropriately chosen chain of Bott-Samelson varieties depends only
on the combinatorics of the Weyl group \(W\) and ultimately we infer
the isomorphism between \(X\) and \(G/B\).
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