

Shafarevich Seminar
August 28, 2018 15:00, Moscow, Steklov Mathematical Institute, room 540 (Gubkina 8)






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 wellknown 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.

