Speaker
Description
The birational Kawamata--Viehweg vanishing theorem (KVV for short) is known
to hold in considerable generality in equal characteristic zero. By contrast,
it fails in general in positive characteristic. Nevertheless, it may still
hold for certain types of birational morphisms.
In his 2008 paper in the Michigan Mathematical Journal, Lipman studied
sequences of blow-ups at closed points over regular schemes, including in
mixed characteristic, and stated a vanishing theorem that we call
"Lipman's goal statement". This can be viewed as a special form of KVV.
Although the argument in that paper is incorrect, this does not imply that
the goal statement itself is false.
In this talk, we prove the goal statement under either of the assumptions,
(1) the schemes are essentially of finite type over a perfect field $k$ of
characteristic $p\geq N-1$, where $N$ is their dimension, or
(2) the schemes have dimension at most $3$.