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Shafarevich Seminar
August 28, 2018 15:00, Moscow, Steklov Mathematical Institute, room 540 (Gubkina 8)
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Boundedness and existence of $n$-complements
V. V. Shokurov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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Abstract:
This talk is about the theorem of
boundedness and existence of
$n$-complements of a local relative
pair with boundar. The morphism of pair
is supposed to be an FT contraction.
The local property means that the morphism
is defined over a neighborhood of (not
necessarily closed) point. For the existence of
$n$-complements it is sufficient the existence of
numerical or $R$-complements. The boundedness
means that for $n$-comlements of pairs of a fixed dimension
it is sufficient a finite set of positive
integers $n$. Moreover, such sets has
some additional properties: divisibility and
aproximation properties for irrational numbers
and vectors. The latter properties implies
important applications to certain questions and
results about acc of some well-known invariants
of log pairs. E.g., acc of the log canonical thresholds.
This allows to give a new more simple proof
for the finite generatedness of log canonical
ring and for the existence of flips.
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